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a/docs/html/img9.png and b/docs/html/img9.png differ diff --git a/docs/html/index.html b/docs/html/index.html index 3655cc3c..3dcd3433 100644 --- a/docs/html/index.html +++ b/docs/html/index.html @@ -23,18 +23,18 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next up previous - contents
- Next: Next: Abstract -   Contents

@@ -77,76 +77,74 @@ Feb. 28, 2017
diff --git a/docs/html/node1.html b/docs/html/node1.html index 62abaca4..cf182508 100644 --- a/docs/html/node1.html +++ b/docs/html/node1.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
- Next: Next: Contents - Up: Up: userhtml - Previous: Previous: userhtml -   Contents

@@ -55,24 +55,22 @@ original version by: Nikos Drakos, CBLU, University of Leeds Abstract -MLD2P4 (MULTI-LEVEL DOMAIN DECOMPOSITION PARALLEL PRECONDITIONERS PACKAGE BASED ON -PSBLAS) is a package of parallel algebraic multi-level preconditioners. -The first release made available various versions of -one-level additive and multi-level additive -and hybrid Schwarz preconditioners. -The package has been extended to include further multi-level cycles and smoothers widely used in -multigrid methods. -In the multi-level case, a purely algebraic approach -is applied to generate coarse-level corrections, so that no geometric background is needed -concerning the matrix to be preconditioned. The matrix is assumed to be square, real -or complex. +MLD2P4 (MULTI-LEVEL DOMAIN DECOMPOSITION PARALLEL PRECONDITIONERS PACKAGE +BASED ON PSBLAS) is a package of parallel algebraic multi-level preconditioners. +The first release of MLD2P4 made available multi-level additive and hybrid Schwarz +preconditioners, as well as one-level additive Schwarz preconditioners. The package +has been extended to include further multi-level cycles and smoothers widely used in +multigrid methods. In the multi-level case, a purely algebraic approach is applied to +generate coarse-level corrections, so that no geometric background is needed +concerning the matrix to be preconditioned. The matrix is assumed to be square, +real or complex.

-MLD2P4 has been designed to provide scalable and easy-to-use preconditioners in the -context of the PSBLAS (Parallel Sparse Basic Linear Algebra Subprograms) +MLD2P4 has been designed to provide scalable and easy-to-use preconditioners +in the context of the PSBLAS (Parallel Sparse Basic Linear Algebra Subprograms) computational framework and can be used in conjuction with the Krylov solvers -available in this framework. MLD2P4 enables the user to easily specify different features -of an algebraic multi-level preconditioner, thus allowing to search +available in this framework. MLD2P4 enables the user to easily specify different +features of an algebraic multi-level preconditioner, thus allowing to search for the ``best'' preconditioner for the problem at hand.

@@ -92,26 +90,26 @@ the user interface of MLD2P4.

- next - up - previous - contents
- Next: Next: Contents - Up: Up: userhtml - Previous: Previous: userhtml -   Contents diff --git a/docs/html/node10.html b/docs/html/node10.html index ad0d9261..a64fde3f 100644 --- a/docs/html/node10.html +++ b/docs/html/node10.html @@ -25,26 +25,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
- Next: Multi-level Domain Decomposition Background - Up: Next: Multigrid Background + Up: Configuring and Building MLD2P4 - Previous: Previous: Bug reporting -   Contents

@@ -62,19 +62,20 @@ both of them are further divided into fileread and
contains a set of simple example programs with a predefined choice of preconditioners, selectable via integer values. These are intended to get an acquaintance with the - multilevel preconditioners. + multi-level preconditioners available in MLD2P4.
tests
contains a set of more sophisticated examples that will allow the user, via the input files in the runs subdirectories, to experiment with the full range of preconditioners - implemented in the library. + implemented in the package.
The fileread directories contain sample programs that read sparse matrices from files, according to the Matrix Market or the -Harwell-Boeing storage format; the pdegen instead generate -matrices in full parallel mode from the discretization of a sample PDE. +Harwell-Boeing storage format; the pdegen programs generate +matrices in full parallel mode from the discretization of a sample partial +differential equation.
diff --git a/docs/html/node11.html b/docs/html/node11.html index feb0b7b0..0b2a7787 100644 --- a/docs/html/node11.html +++ b/docs/html/node11.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Multi-level Domain Decomposition Background - +Multigrid Background + @@ -18,7 +18,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
- Next: Multi-level Schwarz Preconditioners - Up: Next: AMG preconditioners + Up: userhtml - Previous: Previous: Example and test programs -   Contents

@@ -53,126 +53,48 @@ original version by: Nikos Drakos, CBLU, University of Leeds


-Multi-level Domain Decomposition Background -

- -

-Domain Decomposition (DD) preconditioners, coupled with Krylov iterative -solvers, are widely used in the parallel solution of large and sparse linear systems. -These preconditioners are based on the divide and conquer technique: the matrix -to be preconditioned is divided into submatrices, a ``local'' linear system -involving each submatrix is (approximately) solved, and the local solutions are used -to build a preconditioner for the whole original matrix. This process -often corresponds to dividing a physical domain associated to the original matrix -into subdomains, e.g. in a PDE discretization, to (approximately) solving the -subproblems corresponding to the subdomains and to building an approximate -solution of the original problem from the local solutions -[6,7,23]. - -

-Additive Schwarz preconditioners are DD preconditioners using overlapping -submatrices, i.e. with some common rows, to couple the local information -related to the submatrices (see, e.g., [23]). -The main motivation for choosing Additive Schwarz preconditioners is their -intrinsic parallelism. A drawback of these -preconditioners is that the number of iterations of the preconditioned solvers -generally grows with the number of submatrices. This may be a serious limitation -on parallel computers, since the number of submatrices usually matches the number -of available processors. Optimal convergence rates, i.e. iteration numbers -independent of the number of submatrices, can be obtained by correcting the -preconditioner through a suitable approximation of the original linear system -in a coarse space, which globally couples the information related to the single -submatrices. - -

-Two-level Schwarz preconditioners are obtained -by combining basic (one-level) Schwarz preconditioners with a coarse-level -correction. In this context, the one-level preconditioner is often -called `smoother'. Different two-level preconditioners are obtained by varying the -choice of the smoother and of the coarse-level correction, and the -way they are combined [23]. The same reasoning can be applied starting -from the coarse-level system, i.e. a coarse-space correction can be built -from this system, thus obtaining multi-level preconditioners. - -

-It is worth noting that optimal preconditioners do not necessarily correspond -to minimum execution times. Indeed, to obtain effective multi-level preconditioners -a tradeoff between optimality of convergence and the cost of building and applying -the coarse-space corrections must be achieved. The choice of the number of levels, -i.e. of the coarse-space corrections, also affects the effectiveness of the -preconditioners. One more goal is to get convergence rates as less sensitive -as possible to variations in the matrix coefficients. - -

-Two main approaches can be used to build coarse-space corrections. The geometric approach -applies coarsening strategies based on the knowledge of some physical grid associated -to the matrix and requires the user to define grid transfer operators from the fine -to the coarse levels and vice versa. This may result difficult for complex geometries; -furthermore, suitable one-level preconditioners may be required to get efficient -interplay between fine and coarse levels, e.g. when matrices with highly varying coefficients -are considered. The algebraic approach builds coarse-space corrections using only matrix -information. It performs a fully automatic coarsening and enforces the interplay between -the fine and coarse levels by suitably choosing the coarse space and the coarse-to-fine -interpolation [25]. - -

-MLD2P4 uses a pure algebraic approach for building the sequence of coarse matrices -starting from the original matrix. The algebraic approach is based on the smoothed -aggregation algorithm [1,27]. A decoupled version -of this algorithm is implemented, where the smoothed aggregation is applied locally -to each submatrix [26]. In the next two subsections we provide -a brief description of the multi-level Schwarz preconditioners and of the smoothed -aggregation technique as implemented in MLD2P4. For further details the reader -is referred to [2,3,4,8,23]. - -

-

+Multigrid Background + Multigrid preconditioners, coupled with Krylov iterative solvers, are widely used in the parallel solution of large and sparse linear systems, because of their optimality in the solution of linear systems arising from the discretization of scalar elliptic Partial Differential Equations (PDEs) on regular grids. Optimality, also known as algorithmic scalability, is the property of having a computational cost per iteration that depends linearly on the problem size, and a convergence rate that is independent of the problem size. Multigrid preconditioners are based on a recursive application of a two-grid process consisting of smoother iterations and a coarse-space (or coarse-level) correction. The smoothers may be either basic iterative methods, such as the Jacobi and Gauss-Seidel ones, or more complex subspace-correction methods, such as the Schwarz ones. The coarse-space correction consists of solving, in an appropriately chosen coarse space, the residual equation associated with the approximate solution computed by the smoother, and of using the solution of this equation to correct the previous approximation. The transfer of information between the original (fine) space and the coarse one is performed by using suitable restriction and prolongation operators. The construction of the coarse space and the corresponding transfer operators is carried out by applying a so-called coarsening algorithm to the system matrix. Two main approaches can be used to perform coarsening: the geometric approach, which exploits the knowledge of some physical grid associated with the matrix and requires the user to define transfer operators from the fine to the coarse level and vice versa, and the algebraic approach, which builds the coarse-space correction and the associate transfer operators using only matrix information. The first approach may be difficult when the system comes from discretizations on complex geometries; furthermore, ad hoc one-level smoothers may be required to get an efficient interplay between fine and coarse levels, e.g., when matrices with highly varying coefficients are considered. The second approach performs a fully automatic coarsening and enforces the interplay between fine and coarse level by suitably choosing the coarse space and the coarse-to-fine interpolation (see, e.g., [2,27,25] for details.) MLD2P4 uses a pure algebraic approach, based on the smoothed aggregation algorithm [1,29], for building the sequence of coarse matrices and transfer operators, starting from the original one. A decoupled version of this algorithm is implemented, where the smoothed aggregation is applied locally to each submatrix [28]. A brief description of the AMG preconditioners implemented in MLD2P4 is given in Sections 4.1-4.3. For further details the reader is referred to [3,4,5,9]. We note that optimal multigrid preconditioners do not necessarily correspond to minimum execution times in a parallel setting. Indeed, to obtain effective parallel multigrid preconditioners, a tradeoff between the optimality and the cost of building and applying the smoothers and the coarse-space corrections must be achieved. Effective parallel preconditioners require algorithmic scalability to be coupled with implementation scalability, i.e., a computational cost per iteration which remains (almost) constant as the number of parallel processors increases.
Subsections
- next - up - previous - contents
- Next: Multi-level Schwarz Preconditioners - Up: Next: AMG preconditioners + Up: userhtml - Previous: Previous: Example and test programs -   Contents diff --git a/docs/html/node12.html b/docs/html/node12.html index 01fc4aa3..c284507f 100644 --- a/docs/html/node12.html +++ b/docs/html/node12.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Multi-level Schwarz Preconditioners - +AMG preconditioners + @@ -18,7 +18,6 @@ original version by: Nikos Drakos, CBLU, University of Leeds - @@ -26,26 +25,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
- Next: Smoothed Aggregation - Up: Multi-level Domain Decomposition Background - Previous: Multi-level Domain Decomposition Background -   Next: Getting Started + Up: Multigrid Background + Previous: Multigrid Background +   Contents

@@ -53,713 +52,105 @@ original version by: Nikos Drakos, CBLU, University of Leeds


-Multi-level Schwarz Preconditioners -

+AMG preconditioners + In order to describe the AMG preconditioners available in MLD2P4, we consider a linear system
+
-

-The Multilevel preconditioners implemented in MLD2P4 are obtained by combining -AS preconditioners with coarse-space corrections; therefore -we first provide a sketch of the AS preconditioners. - -

-Given the linear system , + + + + +
\begin{displaymath} +Ax=b, +\end{displaymath} +(2)
+

where $A=(a_{ij}) \in \Re^{n \times n}$ is a -nonsingular sparse matrix with a symmetric nonzero pattern, -let $G=(W,E)$ be the adjacency graph of $A=(a_{ij}) \in \mathbb{R}^{n \times n}$ is a nonsingular sparse matrix; for ease of presentation we assume $A$, where -$W=\{1, 2, \ldots, n\}$ -and -$E=\{(i,j) : a_{ij} \neq 0\}$ are the vertex set and the edge set of $G$, -respectively. Two vertices are called adjacent if there is an edge connecting -them. For any integer $\delta > 0$, a $\delta$-overlap -partition of $W$ can be defined recursively as follows. -Given a 0-overlap (or non-overlapping) partition of $W$, -i.e. a set of $m$ disjoint nonempty sets -$W_i^0 \subset W$ such that - -$\cup_{i=1}^m W_i^0 = W$, a $\delta$-overlap -partition of $W$ is obtained by considering the sets - -$W_i^\delta \supset W_i^{\delta-1}$ obtained by including the vertices that -are adjacent to any vertex in -$W_i^{\delta-1}$. - -

-Let $n_i^\delta$ be the size of $W_i^\delta$ and -$R_i^{\delta} \in -\Re^{n_i^\delta \times n}$ the restriction operator that maps -a vector $v \in \Re^n$ onto the vector -$v_i^{\delta} \in \Re^{n_i^\delta}$ -containing the components of $v$ corresponding to the vertices in -$W_i^\delta$. The transpose of $R_i^{\delta}$ is a -prolongation operator from -$\Re^{n_i^\delta}$ to $\Re^n$. -The matrix -$A_i^\delta=R_i^\delta A (R_i^\delta)^T \in -\Re^{n_i^\delta \times n_i^\delta}$ can be considered -as a restriction of $A$ is real, but the results are valid for the complex case as well. Let us assume as finest index space the set of row (column) indices of $A$ corresponding to the set $W_i^{\delta}$. - -

-The classical one-level AS preconditioner is defined by -

-
- - -\begin{displaymath} -M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T -(A_i^\delta)^{-1} R_i^{\delta}, -\end{displaymath} -
-
-

-where $A_i^\delta$ is assumed to be nonsingular. Its application -to a vector $v \in \Re^n$ within a Krylov solver requires the following -three steps: - -
    -
  1. restriction of $v$ as -$v_i = R_i^{\delta} v$, $i=1,\ldots,m$; -
    2. -
  2. solution of the linear systems -$A_i^\delta w_i = v_i$, - $i=1,\ldots,m$; -
    3. -
  3. prolongation and sum of the $w_i$'s, i.e. -$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$. -
    4. -
-Note that the linear systems at step 2 are usually solved approximately, -e.g. using incomplete LU factorizations such as ILU($p$), MILU($p$) and -ILU($p,t$) [22, Chapter 10]. - -

-A variant of the classical AS preconditioner that outperforms it -in terms of convergence rate and of computation and communication -time on parallel distributed-memory computers is the so-called Restricted AS -(RAS) preconditioner [5,15]. It -is obtained by zeroing the components of $w_i$ corresponding to the -overlapping vertices when applying the prolongation. Therefore, -RAS differs from classical AS by the prolongation operators, -which are substituted by -$(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$, -where $\tilde{R}_i^0$ is obtained by zeroing the rows of $R_i^\delta$ -corresponding to the vertices in $W_i^\delta \backslash W_i^0$: -

-
- - -\begin{displaymath} -M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T -(A_i^\delta)^{-1} R_i^{\delta}. -\end{displaymath} -
-
-

-Analogously, the AS variant called AS with Harmonic extension (ASH) -is defined by -

+ WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0" + SRC="img5.png" + ALT="$\Omega = \{1, 2, \ldots, n\}$">. Any algebraic multilevel preconditioners implemented in MLD2P4 generates a hierarchy of index spaces and a corresponding hierarchy of matrices,

\begin{displaymath}M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T -(A_i^\delta)^{-1} \tilde{R}_i^0. -\end{displaymath} + WIDTH="398" HEIGHT="30" BORDER="0" + SRC="img6.png" + ALT="\begin{displaymath}\Omega^1 \equiv \Omega \supset \Omega^2 \supset \ldots \supset \Omega^{nlev}, \quad A^1 \equiv A, A^2, \ldots, A^{nlev}, \end{displaymath}">

-We note that for $\delta=0$ the three variants of the AS preconditioner are -all equal to the block-Jacobi preconditioner. - -

-As already observed, the convergence rate of the one-level Schwarz -preconditioned iterative solvers deteriorates as the number $m$ of partitions -of $W$ increases [7,23]. To reduce the dependency -of the number of iterations on the degree of parallelism we may -introduce a global coupling among the overlapping partitions by defining -a coarse-space approximation $A_C$ of the matrix $A$. -In a pure algebraic setting, $A_C$ is usually built with -the Galerkin approach. Given a set $W_C$ of coarse vertices, -with size $n_C$, and a suitable restriction operator - -$R_C \in \Re^{n_C \times n}$, $A_C$ is defined as -

-
- - -\begin{displaymath} -A_C=R_C A R_C^T -\end{displaymath} -
-
-

-and the coarse-level correction matrix to be combined with a generic -one-level AS preconditioner $M_{1L}$ is obtained as -

-
- - -\begin{displaymath} -M_{C}^{-1}= R_C^T A_C^{-1} R_C, -\end{displaymath} -
-
-

-where $A_C$ is assumed to be nonsingular. The application of $M_{C}^{-1}$ -to a vector $v$ corresponds to a restriction, a solution and -a prolongation step; the solution step, involving the matrix $A_C$, -may be carried out also approximately. - -

-The combination of $M_{C}$ and $M_{1L}$ may be -performed in either an additive or a multiplicative framework. -In the former case, the two-level additive Schwarz preconditioner -is obtained: -

-
- - -\begin{displaymath} -M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}. -\end{displaymath} -
-
-

-Applying $M_{2L-A}^{-1}$ to a vector $v$ within a Krylov solver -corresponds to applying $M_{C}^{-1}$ -and $M_{1L}^{-1}$ to $v$ independently and then summing up -the results. - -

-In the multiplicative case, the combination can be -performed by first applying the smoother $M_{1L}^{-1}$ and then -the coarse-level correction operator $M_{C}^{-1}$: -

-
- - -\begin{displaymath} -\begin{array}{l} -w = M_{1L}^{-1} v, \\ -z = w + M_{C}^{-1} (v-Aw); -\end{array}\end{displaymath} -
-
-

-this corresponds to the following two-level hybrid pre-smoothed -Schwarz preconditioner: -

-
- - -\begin{displaymath} -M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}. -\end{displaymath} -
-
-

-On the other hand, by applying the smoother after the coarse-level correction, -i.e. by computing -

-
- - \begin{displaymath} -\begin{array}{l} -w = M_{C}^{-1} v , \\ -z = w + M_{1L}^{-1} (v-Aw) , -\end{array}\end{displaymath} -
-
-

-the two-level hybrid post-smoothed -Schwarz preconditioner is obtained: -

+ WIDTH="34" HEIGHT="15" ALIGN="BOTTOM" BORDER="0" + SRC="img7.png" + ALT="$\mathbb{R}^{n_{k}}$"> is associated with $\Omega^k$, where $n_k$ is the size of $\Omega^k$. For all $k < nlev$, a restriction operator and a prolongation one are built, which connect two levels $k$ and $k+1$:

\begin{displaymath} -M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}. -\end{displaymath} + WIDTH="255" HEIGHT="30" BORDER="0" + SRC="img13.png" + ALT="\begin{displaymath} P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad R^k \in \mathbb{R}^{n_{k+1}\times n_k}; \end{displaymath}">

-

-One more variant of two-level hybrid preconditioner is obtained by applying -the smoother before and after the coarse-level correction. In this case, the -preconditioner is symmetric if $A$, $M_{1L}$ and $M_{C}$ are symmetric. - -

-As previously noted, on parallel computers the number of submatrices usually matches -the number of available processors. When the size of the system to be preconditioned -is very large, the use of many processors, i.e. of many small submatrices, often -leads to a large coarse-level system, whose solution may be computationally expensive. -On the other hand, the use of few processors often leads to local sumatrices that -are too expensive to be processed on single processors, because of memory and/or -computing requirements. Therefore, it seems natural to use a recursive approach, -in which the coarse-level correction is re-applied starting from the current -coarse-level system. The corresponding preconditioners, called multi-level -preconditioners, can significantly reduce the computational cost of preconditioning -with respect to the two-level case (see [23, Chapter 3]). -Additive and hybrid multilevel preconditioners -are obtained as direct extensions of the two-level counterparts. -For a detailed descrition of them, the reader is -referred to [23, Chapter 3]. -The algorithm for the application of a multi-level hybrid -post-smoothed preconditioner $M$ to a vector $v$, i.e. for the -computation of $w=M^{-1}v$, is reported, for -example, in Figure 1. Here the number of levels -is denoted by $nlev$ and the levels are numbered in increasing order starting -from the finest one, i.e. the finest level is level 1; the coarse matrix -and the corresponding basic preconditioner at each level $l$ are denoted by $A_l$ and -$M_l$, respectively, with $A_1=A$, while the related restriction operator is -denoted by $R_l$. - -

- - - -
Figure 1: -Application of the multi-level hybrid post-smoothed preconditioner.
-
- -\framebox{ -\begin{minipage}{.85\textwidth} {\small -\begin{tabbing} -\quad \=\quad... -...= y_l+r_l$\\ -\textbf{endfor} \\ [1mm] -$w = y_1$; -\end{tabbing}} -\end{minipage}} - -
-
- -

-

- - -next - -up - -previous - -contents -
- Next: Smoothed Aggregation - Up: Multi-level Domain Decomposition Background - Previous: Multi-level Domain Decomposition Background -   Contents - +

+
diff --git a/docs/html/node13.html b/docs/html/node13.html index 98fab665..1229c8ec 100644 --- a/docs/html/node13.html +++ b/docs/html/node13.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Smoothed Aggregation - +Getting Started + @@ -18,296 +18,206 @@ original version by: Nikos Drakos, CBLU, University of Leeds - - + + + - next - -up +up + previous - contents
- Next: Getting Started - Up: Multi-level Domain Decomposition Background - Previous: Multi-level Schwarz Preconditioners -   Next: Examples + Up: userhtml + Previous: AMG preconditioners +   Contents

-

+


-Smoothed Aggregation -

+Getting Started +

-In order to define the restriction operator $R_C$, which is used to compute -the coarse-level matrix $A_C$, MLD2P4 uses the smoothed aggregation -algorithm described in [1,27]. -The basic idea of this algorithm is to build a coarse set of vertices -$W_C$ by suitably grouping the vertices of $W$ into disjoint subsets -(aggregates), and to define the coarse-to-fine space transfer operator $R_C^T$ by -applying a suitable smoother to a simple piecewise constant -prolongation operator, to improve the quality of the coarse-space correction. - -

-Three main steps can be identified in the smoothed aggregation procedure: +We describe the basics for building and applying MLD2P4 one-level and multi-level +(i.e., AMG) preconditioners with the Krylov solvers included in PSBLAS [17]. +The following steps are required:

    -
  1. coarsening of the vertex set $W$, to obtain $W_C$; +
  2. Declare the preconditioner data structure. It is a derived data type, + mld_xprec_ type, where x may be s, d, c + or z, according to the basic data type of the sparse matrix + (s = real single precision; d = real double precision; + c = complex single precision; z = complex double precision). + This data structure is accessed by the user only through the MLD2P4 routines, + following an object-oriented approach. +
    3. +
  3. Allocate and initialize the preconditioner data structure, according to + a preconditioner type chosen by the user. This is performed by the routine + init, which also sets defaults for each preconditioner + type selected by the user. The preconditioner types and the defaults associated + with them are given in Table 1, where the strings used by + init to identify the preconditioner types are also given. + Note that these strings are valid also if uppercase letters are substituted by + corresponding lowercase ones.
    4. -
  4. construction of the prolongator $R_C^T$; +
  5. Modify the selected preconditioner type, by properly setting + preconditioner parameters. This is performed by the routine set. + This routine must be called only if the user wants to modify the default values + of the parameters associated with the selected preconditioner type, to obtain a variant + of that preconditioner. Examples of use of set are given in + Section 5.1; a complete list of all the + preconditioner parameters and their allowed and default values is provided in + Section 6, Tables 2-8.
    6. -
  6. application of $R_C$ and $R_C^T$ to build $A_C$. +
  7. Build the preconditioner for a given matrix. If the selected preconditioner + is multi-level, then two steps must be performed, as specified next. +
    +
    4.1
    +
    Build the aggregation hierarchy for a given matrix. This is +performed by the routine hierarchy_build. +
    +
    4.2
    +
    Build the preconditioner for a given matrix. This is performed +by the routine smoothers_build. +
    +
    + If the selected preconditioner is one-level, it is built in a single step, +performed by the routine bld. +
    8. +
  8. Apply the preconditioner at each iteration of a Krylov solver. + This is performed by the routine aply. When using the PSBLAS Krylov solvers, + this step is completely transparent to the user, since aply is called + by the PSBLAS routine implementing the Krylov solver (psb_krylov). +
    9. +
  9. Free the preconditioner data structure. This is performed by + the routine free. This step is complementary to step 1 and should + be performed when the preconditioner is no more used.
-

-To perform the coarsening step, we have implemented the aggregation algorithm sketched -in [4]. According to [27], a modification of -this algorithm has been actually considered, -in which each aggregate $N_r$ is made of vertices of $W$ that are strongly coupled -to a certain root vertex $r \in W$, i.e.

-
- - -\begin{displaymath}N_r = \left\{s \in W: \vert a_{rs}\vert > \theta \sqrt{\vert a_{rr}a_{ss}\vert} \right\} -\cup \left\{ r \right\} , -\end{displaymath} -
-
-

-for a given -$\theta \in [0,1]$. -Since this algorithm has a sequential nature, a decoupled version of -it has been chosen, where each processor $i$ independently applies the algorithm to -the set of vertices $W_i^0$ assigned to it in the initial data distribution. This -version is embarrassingly parallel, since it does not require any data communication. -On the other hand, it may produce non-uniform aggregates near boundary vertices, -i.e. near vertices adjacent to vertices in other processors, and is strongly -dependent on the number of processors and on the initial partitioning of the matrix $A$. -Nevertheless, this algorithm has been chosen for the implementation in MLD2P4, -since it has been shown to produce good results in practice -[3,4,26].

-The prolongator $P_C=R_C^T$ is built starting from a tentative prolongator - -$P \in \Re^{n \times n_C}$, defined as -
-

+All the previous routines are available as methods of the preconditioner object. +A detailed description of them is given in Section 6. +Examples showing the basic use of MLD2P4 are reported in Section 5.1. - - - - +

+

+
+
\begin{displaymath} -P=(p_{ij}), \quad p_{ij}= -\left\{ \begin{array}{ll} -1 & \qu... -...\in V^j_C \\ -0 & \quad \mbox{otherwise} -\end{array} \right. . -\end{displaymath} -(2)
+ + + HREF="node27.html#PSBLASGUIDE">17]. @@ -89,56 +90,20 @@ made by the user through the routines init and set + HREF="node27.html#PSBLASGUIDE">17]. - +
Table 1: +Preconditioner types, corresponding strings and default choices. +
+
+ + + + + + + + + + + + + + + + + + + + + + + + +
TYPESTRINGDEFAULT PRECONDITIONER
No preconditioner'NOPREC'Considered only to use the PSBLAS + Krylov solvers with no preconditioner.
Diagonal'DIAG' or 'JACOBI'Diagonal preconditioner. + For any zero diagonal entry of the matrix to be preconditioned, + the corresponding entry of he preconditioner is set to 1.
Block Jacobi'BJAC'Block-Jacobi with ILU(0) on the local blocks.
Additive Schwarz'AS'Restricted Additive Schwarz (RAS), + with overlap 1 and ILU(0) on the local blocks.
Multilevel'ML'V-cycle with one hybrid forward Gauss-Seidel + (GS) sweep as pre-smoother and one hybrid backward + GS sweep as post-smoother, basic smoothed aggregation + as coarsening algorithm, and LU (plus triangular solve) + as coarsest-level solver. See the default values in + Tables 2-8 + for further details of the preconditioner.
-

-$P_C$ is obtained by -applying to $P$ a smoother -$S \in \Re^{n \times n}$: -
-
- - - - - +
\begin{displaymath} -P_C = S P, -\end{displaymath} -(3)
-

-in order to remove oscillatory components from the range of the prolongator -and hence to improve the convergence properties of the multi-level -Schwarz method [1,25]. -A simple choice for $S$ is the damped Jacobi smoother: +


-
- - - - -
\begin{displaymath} -S = I - \omega D^{-1} A , -\end{displaymath} -(4)
-

-where the value of $\omega$ can be chosen -using some estimate of the spectral radius of $D^{-1}A$ [1].

+Note that the module mld_prec_mod, containing the definition of the +preconditioner data type and the interfaces to the routines of MLD2P4, +must be used in any program calling such routines. +The modules psb_base_mod, for the sparse matrix and communication descriptor +data types, and psb_krylov_mod, for interfacing with the +Krylov solvers, must be also used (see Section 5.1). +
+

+Remark 1. Coarsest-level solvers based on the LU factorization, +such as those implemented in UMFPACK, MUMPS, SuperLU, and SuperLU_Dist, +usually lead to smaller numbers of preconditioned Krylov +iterations than inexact solvers, when the linear system comes from +a standard discretization of basic scalar elliptic PDE problems. However, +this does not necessarily correspond to the smallest execution time +on parallel computers. +

+

+ +Subsections + + +
- next - -up +up + previous - contents
- Next: Getting Started - Up: Multi-level Domain Decomposition Background - Previous: Multi-level Schwarz Preconditioners -   Next: Examples + Up: userhtml + Previous: AMG preconditioners +   Contents diff --git a/docs/html/node14.html b/docs/html/node14.html index b8317669..6ecc3856 100644 --- a/docs/html/node14.html +++ b/docs/html/node14.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Getting Started - +Examples + @@ -18,206 +18,302 @@ original version by: Nikos Drakos, CBLU, University of Leeds - - - + + - next - + up - previous - contents
- Next: Examples - Up: userhtml - Previous: Smoothed Aggregation -   Next: User Interface + Up: Getting Started + Previous: Getting Started +   Contents

-

+


-Getting Started -

+Examples + + +

+The code reported in Figure 2 shows how to set and apply the default +multi-level preconditioner available in the real double precision version +of MLD2P4 (see Table 1). This preconditioner is chosen +by simply specifying 'ML' as the second argument of P%init +(a call to P%set is not needed) and is applied with the CG +solver provided by PSBLAS (the matrix of the system to be solved is +assumed to be positive definite). As previously observed, the modules +psb_base_mod, mld_prec_mod and psb_krylov_mod +must be used by the example program.

-We describe the basics for building and applying MLD2P4 one-level and multi-level -(i.e., AMG) preconditioners with the Krylov solvers included in PSBLAS [16]. -The following steps are required: - -

    -
  1. Declare the preconditioner data structure. It is a derived data type, - mld_xprec_ type, where x may be s, d, c - or z, according to the basic data type of the sparse matrix - (s = real single precision; d = real double precision; - c = complex single precision; z = complex double precision). - This data structure is accessed by the user only through the MLD2P4 routines, - following an object-oriented approach. -
    2. -
  2. Allocate and initialize the preconditioner data structure, according to - a preconditioner type chosen by the user. This is performed by the routine - init, which also sets defaults for each preconditioner - type selected by the user. The preconditioner types and the defaults associated - with them are given in Table 1, where the strings used by - init to identify the preconditioner types are also given. - Note that these strings are valid also if uppercase letters are substituted by - corresponding lowercase ones. -
    3. -
  3. Modify the selected preconditioner type, by properly setting - preconditioner parameters. This is performed by the routine set. - This routine must be called only if the user wants to modify the default values - of the parameters associated with the selected preconditioner type, to obtain a variant - of that preconditioner. Examples of use of set are given in - Section 5.1; a complete list of all the - preconditioner parameters and their allowed and default values is provided in - Section 6, Tables 2-8. -
    4. -
  4. Build the preconditioner for a given matrix. If the selected preconditioner - is multi-level, then two steps must be performed, as specified next. -
    -
    4.1
    -
    Build the aggregation hierarchy for a given matrix. This is -performed by the routine hierarchy_bld. -
    -
    4.2
    -
    Build the preconditioner for a given matrix. This is performed -by the routine smoothers_bld. -
    -
    - If the selected preconditioner is one-level, it is built in a single step, -performed by the routine bld. -
    5. -
  5. Apply the preconditioner at each iteration of a Krylov solver. - This is performed by the routine aply. When using the PSBLAS Krylov solvers, - this step is completely transparent to the user, since aply is called - by the PSBLAS routine implementing the Krylov solver (psb_krylov). -
    6. -
  6. Free the preconditioner data structure. This is performed by - the routine free. This step is complementary to step 1 and should - be performed when the preconditioner is no more used. -
    7. -
+The part of the code concerning the +reading and assembling of the sparse matrix and the right-hand side vector, performed +through the PSBLAS routines for sparse matrix and vector management, is not reported +here for brevity; the statements concerning the deallocation of the PSBLAS +data structure are neglected too. +The complete code can be found in the example program file mld_dexample_ml.f90, +in the directory examples/fileread of the MLD2P4 implementation (see +Section 3.5). A sample test problem along with the relevant +input data is available in examples/fileread/runs. +For details on the use of the PSBLAS routines, see the PSBLAS User's +Guide [17].

-All the previous routines are available as methods of the preconditioner object. -A detailed description of them is given in Section 6. -Examples showing the basic use of MLD2P4 are reported in Section 5.1. +The setup and application of the default multi-level preconditioner +for the real single precision and the complex, single and double +precision, versions are obtained with straightforward modifications of the previous +example (see Section 6 for details). If these versions are installed, +the corresponding codes are available in examples/fileread/.

-

-
+ +
-
Table 1: -Preconditioner types, corresponding strings and default choices. +Figure 2: +setup and application of the default multi-level preconditioner (example 1).
- - - - - - - - - - - - - - - - - - - - - - - - - +
TYPESTRINGDEFAULT PRECONDITIONER
No preconditioner'NOPREC'Considered only to use the PSBLAS - Krylov solvers with no preconditioner.
Diagonal'DIAG' or 'JACOBI'Diagonal preconditioner. - For any zero diagonal entry of the matrix to be preconditioned, - the corresponding entry of he preconditioner is set to 1.
Block Jacobi'BJAC'Block-Jacobi with ILU(0) on the local blocks.
Additive Schwarz'AS'Restricted Additive Schwarz (RAS), - with overlap 1 and ILU(0) on the local blocks.
Multilevel'ML'V-cycle with one hybrid forward Gauss-Seidel - (GS) sweep as pre-smoother and one hybrid backward - GS sweep as post-smoother, basic smoothed aggregation - as coarsening algorithm, and LU (plus triangular solve) - as coarsest-level solver. See the default values in - Tables 2-8 - for further details of the preconditioner.
+
+
+  use psb_base_mod
+  use mld_prec_mod
+  use psb_krylov_mod
+... ...
+!
+! sparse matrix
+  type(psb_dspmat_type) :: A
+! sparse matrix descriptor
+  type(psb_desc_type)   :: desc_A
+! preconditioner
+  type(mld_dprec_type)  :: P
+! right-hand side and solution vectors
+  type(psb_d_vect_type) :: b, x
+... ...
+!
+! initialize the parallel environment
+  call psb_init(ictxt)
+  call psb_info(ictxt,iam,np)
+... ...
+!
+! read and assemble the spd matrix A and the right-hand side b 
+! using PSBLAS routines for sparse matrix / vector management
+... ...
+!
+! initialize the default multi-level preconditioner, i.e. V-cycle
+! with basic smoothed aggregation, 1 hybrid forward/backward
+! GS sweep as pre/post-smoother and UMFPACK as coarsest-level
+! solver
+  call P%init(P,'ML',info)
+!
+! build the preconditioner
+  call P%hierarchy_build(A,desc_A,P,info)
+  call P%smoothers_build(A,desc_A,P,info)
+
+!
+! set the solver parameters and the initial guess
+  ... ...
+!
+! solve Ax=b with preconditioned CG
+  call psb_krylov('CG',A,P,b,x,tol,desc_A,info)
+  ... ...
+!
+! deallocate the preconditioner
+  call P%free(P,info)
+!
+! deallocate other data structures
+  ... ...
+!
+! exit the parallel environment
+  call psb_exit(ictxt)
+  stop
+
+
+
+
-

-
+

-Note that the module mld_prec_mod, containing the definition of the -preconditioner data type and the interfaces to the routines of MLD2P4, -must be used in any program calling such routines. -The modules psb_base_mod, for the sparse matrix and communication descriptor -data types, and psb_krylov_mod, for interfacing with the -Krylov solvers, must be also used (see Section 5.1). -
+Different versions of the multi-level preconditioner can be obtained by changing +the default values of the preconditioner parameters. The code reported in +Figure 3 shows how to set a V-cycle preconditioner +which applies 1 block-Jacobi sweep as pre- and post-smoother, +and solves the coarsest-level system with 8 block-Jacobi sweeps. +Note that the ILU(0) factorization (plus triangular solve) is used as +local solver for the block-Jacobi sweeps, since this is the default associated +with block-Jacobi and set by P%init. +Furthermore, specifying block-Jacobi as coarsest-level +solver implies that the coarsest-level matrix is distributed +among the processes. +Figure 4 shows how to set a W-cycle preconditioner which +applies no pre-smoother and 2 Gauss-Seidel sweeps as post-smoother, +and solves the coarsest-level system with the multifrontal LU factorization +implemented in MUMPS. It is specified that the coarsest-level +matrix is distributed, since MUMPS can be used on both +replicated and distributed matrices, and by default +it is used on replicated ones. Note the use of the parameter pos +to specify a property only for the pre-smoother or the post-smoother +(see Section 6.2 for more details). +Note also that a Krylov method different from CG must be used to solve +the preconditioned system, since the preconditione in nonsymmetric. +The code fragments shown in Figures 3 and 4 are +included in the example program file mld_dexample_ml.f90 too. + +

+Finally, Figure 5 shows the setup of a one-level +additive Schwarz preconditioner, i.e., RAS with overlap 2. The +corresponding example program is available in the file +mld_dexample_1lev.f90. + +

+For all the previous preconditioners, example programs where the sparse matrix and +the right-hand side are generated by discretizing a PDE with Dirichlet +boundary conditions are also available in the directory examples/pdegen. + +

+ +

+ + + +
Figure 3: +setup of a multi-level preconditioner
+
+
+ +
+
+... ...
+! build a V-cycle preconditioner with 1 block-Jacobi sweep (with 
+! ILU(0) on the blocks) as pre- and post-smoother, and 8  block-Jacobi
+! sweeps (with ILU(0) on the blocks) as coarsest-level solver
+  call P%init(P,'ML',info)
+  call_P%set(P,'SMOOTHER_TYPE','BJAC',info)
+  call P%set(P,'COARSE_SOLVE','BJAC',info)
+  call P%set(P,'COARSE_SWEEPS',8,info)
+  call P%hierarchy_build(A,desc_A,P,info)
+  call P%smoothers_build(A,desc_A,P,info)
+... ...
+
+
+
+
+

+

+
+
+ +

+ +

+ + + +
Figure 4: +setup of a multi-level preconditioner
+
+
+ +
+
+... ...
+! build a W-cycle preconditioner with 2 Gauss-Seidel sweeps as 
+! post-smoother (and no pre-smoother), a distributed coarsest
+! matrix, and MUMPS as coarsest-level solver
+  call P%init(P,'ML',info)
+  call P%set('ML_TYPE','WCYCLE',info)
+  call P%set('SMOOTHER_TYPE','GS',info)
+  call P%set('SMOOTHER_SWEEPS',0,info,pos='PRE')
+  call P%set('SMOOTHER_SWEEPS',2,info,pos='POST')
+  call P%set('COARSE_SOLVE','MUMPS',info)
+  call P%set('COARSE_MAT','DIST',info)
+  call P%hierarchy_build(A,desc_A,P,info)
+  call P%smoothers_build(A,desc_A,P,info)
+... ...
+! solve Ax=b with preconditioned CG
+  call psb_krylov('BICGSTAB',A,P,b,x,tol,desc_A,info)
+
+
+
+ +
+
+

-Remark 1. Coarsest-level solvers based on the LU factorization, -such as those implemented in UMFPACK, MUMPS, SuperLU, and SuperLU_Dist, -usually lead to smaller numbers of preconditioned Krylov -iterations than inexact solvers, when the linear system comes from -a standard discretization of basic scalar elliptic PDE problems. However, -this does not necessarily correspond to the smallest execution time -on parallel computers. + +

+ + + +
Figure 5: +setup of a one-level Schwarz preconditioner.
+
+
+ +
+
+... ...
+! set RAS with overlap 2 and ILU(0) on the local blocks
+  call P%init(P,'AS',info)
+  call P%set(P,'SUB_OVR',2,info)
+  call P%bld(A,desc_A,P,info)
+... ...
+
+
+
+ +
+
+

-

- -Subsections - - -
- next - + up - previous - contents
- Next: Examples - Up: userhtml - Previous: Smoothed Aggregation -   Next: User Interface + Up: Getting Started + Previous: Getting Started +   Contents diff --git a/docs/html/node15.html b/docs/html/node15.html index a8dabbb2..f56a15b5 100644 --- a/docs/html/node15.html +++ b/docs/html/node15.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Examples - +User Interface + @@ -18,302 +18,143 @@ original version by: Nikos Drakos, CBLU, University of Leeds - - + + + - next - -up +up + previous - contents
- Next: User Interface - Up: Getting Started - Previous: Getting Started -   Next: Subroutine init + Up: userhtml + Previous: Examples +   Contents

-

+


-Examples -

+User Interface +

-The code reported in Figure 2 shows how to set and apply the default -multi-level preconditioner available in the real double precision version -of MLD2P4 (see Table 1). This preconditioner is chosen -by simply specifying 'ML' as second argument of P%init -(a call to P%set is not needed) and is applied with the CG -solver provided by PSBLAS (the matrix of the system to be solved is -assumed to be positive definite). As previously observed, the modules -psb_base_mod, mld_prec_mod and psb_krylov_mod -must be used by the example program. +The basic user interface of MLD2P4 consists of eight routines. The six +routines init, set, +hierarchy_build, smoothers_build, +bld, and apply encapsulate all the +functionalities for the setup and the application of any multi-level and one-level +preconditioner implemented in the package. +The routine free deallocates the preconditioner data structure, while +descr prints a description of the preconditioner setup by the user.

-The part of the code concerning the -reading and assembling of the sparse matrix and the right-hand side vector, performed -through the PSBLAS routines for sparse matrix and vector management, is not reported -here for brevity; the statements concerning the deallocation of the PSBLAS -data structure are neglected too. -The complete code can be found in the example program file mld_dexample_ml.f90, -in the directory examples/fileread of the MLD2P4 implementation (see -Section 3.5). A sample test problem along with the relevant -input data is available in examples/fileread/runs. -For details on the use of the PSBLAS routines, see the PSBLAS User's -Guide [16]. +All the routines are available as methods of the preconditioner object. +For each routine, the same user interface is overloaded with +respect to the real/ complex case and the single/double precision; +arguments with appropriate data types must be passed to the routine, +i.e., + +

    +
  • the sparse matrix data structure, containing the matrix to be + preconditioned, must be of type psb_xspmat_type + with x = s for real single precision, x = d + for real double precision, x = c for complex single precision, + x = z for complex double precision; +
    • +
  • the preconditioner data structure must be of type + mld_xprec_type, with x = + s, d, c, z, according to the sparse + matrix data structure; +
    • +
  • the arrays containing the vectors $v$ and $w$ involved in + the preconditioner application $w=M^{-1}v$ must be of type + psb_xvect_type with x = + s, d, c, z, in a manner completely + analogous to the sparse matrix type; +
    • +
  • real parameters defining the preconditioner must be declared + according to the precision of the sparse matrix and preconditioner + data structures (see Section 6.2). +
    • +
+A description of each routine is given in the remainder of this section.

-The setup and application of the default multi-level preconditioner -for the real single precision and the complex, single and double -precision, versions are obtained with straightforward modifications of the previous -example (see Section 6 for details). If these versions are installed, -the corresponding codes are available in examples/fileread/. - -

- -

- - - -
Figure 2: -setup and application of the default multi-level preconditioner (example 1). -
-
-
- -
-
-  use psb_base_mod
-  use mld_prec_mod
-  use psb_krylov_mod
-... ...
-!
-! sparse matrix
-  type(psb_dspmat_type) :: A
-! sparse matrix descriptor
-  type(psb_desc_type)   :: desc_A
-! preconditioner
-  type(mld_dprec_type)  :: P
-! right-hand side and solution vectors
-  type(psb_d_vect_type) :: b, x
-... ...
-!
-! initialize the parallel environment
-  call psb_init(ictxt)
-  call psb_info(ictxt,iam,np)
-... ...
-!
-! read and assemble the spd matrix A and the right-hand side b 
-! using PSBLAS routines for sparse matrix / vector management
-... ...
-!
-! initialize the default multi-level preconditioner, i.e. V-cycle
-! with basic smoothed aggregation, 1 hybrid forward/backward
-! GS sweep as pre/post-smoother and UMFPACK as coarsest-level
-! solver
-  call P%init(P,'ML',info)
-!
-! build the preconditioner
-  call P%hierarchy_bld(A,desc_A,P,info)
-  call P%smoothers_bld(A,desc_A,P,info)
-
-!
-! set the solver parameters and the initial guess
-  ... ...
-!
-! solve Ax=b with preconditioned CG
-  call psb_krylov('CG',A,P,b,x,tol,desc_A,info)
-  ... ...
-!
-! deallocate the preconditioner
-  call P%free(P,info)
-!
-! deallocate other data structures
-  ... ...
-!
-! exit the parallel environment
-  call psb_exit(ictxt)
-  stop
-
-
-
- -
-
- -

-Different versions of the multi-level preconditioner can be obtained by changing -the default values of the preconditioner parameters. The code reported in -Figure 3 shows how to set a V-cycle preconditioner -which applies 1 block-Jacobi sweep as pre- and post-smoother, -and solves the coarsest-level system with 8 block-Jacobi sweeps. -Note that the ILU(0) factorization (plus triangular solve) is used as -local solver for the block-Jacobi sweeps, since this is the default associated -with block-Jacobi and set by P%init. -Furthermore, specifying block-Jacobi as coarsest-level -solver implies that the coarsest-level matrix is distributed -among the processes. -Figure 4 shows how to set a W-cycle preconditioner which -applies no pre-smoother and 2 Gauss-Seidel sweeps as post-smoother, -and solves the coarsest-level system with the multifrontal LU factorization -implemented in MUMPS. It is specified that the coarsest-level -matrix is distributed, since MUMPS can be used on both -replicated and distributed matrices, and by default -it is used on replicated ones. Note the use of the parameter pos -to specify a property only for the pre-smoother or the post-smoother -(see Section 6.2 for more details). -Note also that a Krylov method different from CG must be used to solve -the preconditioned system, since the preconditione in nonsymmetric. -The code fragments shown in Figures 3 and 4 are -included in the example program file mld_dexample_ml.f90 too. - -

-Finally, Figure 5 shows the setup of a one-level -additive Schwarz preconditioner, i.e., RAS with overlap 2. The -corresponding example program is available in the file -mld_dexample_1lev.f90. - -

-For all the previous preconditioners, example programs where the sparse matrix and -the right-hand side are generated by discretizing a PDE with Dirichlet -boundary conditions are also available in the directory examples/pdegen. - -

- -

- - - -
Figure 3: -setup of a multi-level preconditioner
-
-
- -
-
-... ...
-! build a V-cycle preconditioner with 1 block-Jacobi sweep (with 
-! ILU(0) on the blocks) as pre- and post-smoother, and 8  block-Jacobi
-! sweeps (with ILU(0) on the blocks) as coarsest-level solver
-  call P%init(P,'ML',info)
-  call_P%set(P,'SMOOTHER_TYPE','BJAC',info)
-  call P%set(P,'COARSE_SOLVE','BJAC',info)
-  call P%set(P,'COARSE_SWEEPS',8,info)
-  call P%hierarchy_bld(A,desc_A,P,info)
-  call P%smoothers_bld(A,desc_A,P,info)
-... ...
-
-
-
-
-

-

-
-
- -

- -

- - - -
Figure 4: -setup of a multi-level preconditioner
-
-
- -
-
-... ...
-! build a W-cycle preconditioner with 2 Gauss-Seidel sweeps as 
-! post-smoother (and no pre-smoother), a distributed coarsest
-! matrix, and MUMPS as coarsest-level solver
-  call P%init(P,'ML',info)
-  call P%set('ML_TYPE','WCYCLE',info)
-  call P%set('SMOOTHER_TYPE','GS',info)
-  call P%set('SMOOTHER_SWEEPS',0,info,pos='PRE')
-  call P%set('SMOOTHER_SWEEPS',2,info,pos='POST')
-  call P%set('COARSE_SOLVE','MUMPS',info)
-  call P%set('COARSE_MAT','DIST',info)
-  call P%hierarchy_bld(A,desc_A,P,info)
-  call P%smoothers_bld(A,desc_A,P,info)
-... ...
-! solve Ax=b with preconditioned CG
-  call psb_krylov('BICGSTAB',A,P,b,x,tol,desc_A,info)
-
-
-
- -
-
- -

- -

- - - -
Figure 5: -setup of a one-level Schwarz preconditioner.
-
-
- -
-
-... ...
-! set RAS with overlap 2 and ILU(0) on the local blocks
-  call P%init(P,'AS',info)
-  call P%set(P,'SUB_OVR',2,info)
-  call P%bld(A,desc_A,P,info)
-... ...
-
-
-
- -
-

+

+ +Subsections + + +
- next - -up +up + previous - contents
- Next: User Interface - Up: Getting Started - Previous: Getting Started -   Next: Subroutine init + Up: userhtml + Previous: Examples +   Contents diff --git a/docs/html/node16.html b/docs/html/node16.html index 94a235ba..cd013670 100644 --- a/docs/html/node16.html +++ b/docs/html/node16.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -User Interface - +Subroutine init + @@ -18,145 +18,92 @@ original version by: Nikos Drakos, CBLU, University of Leeds - - - + + + - next - + up - previous - contents
- Next: Subroutine init - Up: userhtml - Previous: Examples -   Next: Subroutine set + Up: User Interface + Previous: User Interface +   Contents

-

+


-User Interface -

+Subroutine init + + +

+

+call p%init(ptype,info) + +
+ +

+This routine allocates and initializes the preconditioner +p, according to the preconditioner type chosen by the user. + +

+Arguments +

+ +

+ + + + + + + + + + + + + + + + +
ptypecharacter(len=*), intent(in).
 The type of preconditioner. Its values are specified + in Table 1.
 Note that the strings are case insensitive.
infointeger, intent(out).
 Error code. If no error, 0 is returned. See Section 8 for details.

-The basic user interface of MLD2P4 consists of eight routines. The six -routines init, set, -hierarchy_bld, smoothers_bld, -bld, and apply encapsulate all the -functionalities for the setup and the application of any multi-level and one-level -preconditioner implemented in the package. -The routine free deallocates the preconditioner data structure, while -descr prints a description of the preconditioner setup by the user. +For compatibility with the previous versions of MLD2P4, this routine can be also invoked +as follows:

-All the routines are available as methods of the preconditioner object. -For each routine, the same user interface is overloaded with -respect to the real/ complex case and the single/double precision; -arguments with appropriate data types must be passed to the routine, -i.e., - -

    -
  • the sparse matrix data structure, containing the matrix to be - preconditioned, must be of type psb_xspmat_type - with x = s for real single precision, x = d - for real double precision, x = c for complex single precision, - x = z for complex double precision; -
    • -
  • the preconditioner data structure must be of type - mld_xprec_type, with x = - s, d, c, z, according to the sparse - matrix data structure; -
    • -
  • the arrays containing the vectors $v$ and $w$ involved in - the preconditioner application $w=M^{-1}v$ must be of type - psb_xvect_type with x = - s, d, c, z, in a manner completely - analogous to the sparse matrix type; -
    • -
  • real parameters defining the preconditioner must be declared - according to the precision of the sparse matrix and preconditioner - data structures (see Section 6.2). -
    • -
-A description of each routine is given in the remainder of this section. +
+call mld_precinit(p,ptype,info) + +


- -Subsections - - - -
- - -next - -up - -previous - -contents -
- Next: Subroutine init - Up: userhtml - Previous: Examples -   Contents - diff --git a/docs/html/node17.html b/docs/html/node17.html index 0621ace6..84932d92 100644 --- a/docs/html/node17.html +++ b/docs/html/node17.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Subroutine init - +Subroutine set + @@ -20,51 +20,52 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + - next - + up - previous - contents
- Next: Subroutine set - Up: User Interface - Previous: User Interface -   Next: Subroutine build + Up: User Interface + Previous: Subroutine init +   Contents

-

+


-Subroutine init +Subroutine set

-call p%init(ptype,info) +call p%set(what,val,info [,ilev, ilmax, pos])

-This routine allocates and initializes the preconditioner -p, according to the preconditioner type chosen by the user. +This routine sets the parameters defining the preconditioner p. More +precisely, the parameter identified by what is assigned the value +contained in val.

Arguments @@ -72,21 +73,67 @@ This routine allocates and initializes the preconditioner

- - + + - + + + + - + - + + + + + + + + + + + + + + + + + + +
ptypecharacter(len=*), intent(in).
whatinteger, intent(in) or character(len=*).
 The type of preconditioner. Its values are specified - in Table 1.The parameter to be set. It can be specified by + a predefined constant, or through its name; the string + is case-insensitive. See also + Tables 2-8.
val integer or character(len=*) or + real(psb_spk_) or real(psb_dpk_), + intent(in).
 Note that the strings are case insensitive.The value of the parameter to be set. The list of allowed + values and the corresponding data types is given in + Tables 2-8. + When the value is of type character(len=*), + it is also treated as case insensitive.
info integer, intent(out).
 Error code. If no error, 0 is returned. See Section 8 for details.Error code. If no error, 0 is returned. See Section 8 + for details.
ilevinteger, optional, intent(in).
 For the multi-level preconditioner, the level at which the + preconditioner parameter has to be set. + The levels are numbered in increasing + order starting from the finest one, i.e., level 1 is the finest level. + If ilev is not present, the parameter identified by what + is set at all the appropriate levels (see + Tables 2-8).
ilmaxinteger, optional, intent(in).
 For the multi-level preconditioner, when both + ilev and ilmax are present, the settings + are applied at all levels ilev:ilmax. When + ilev is present but ilmax is not, then + the default is ilmax=ilev. + The levels are numbered in increasing + order starting from the finest one, i.e., level 1 is the finest level.
poscharater(len=*), optional, intent(in).
 Whether the other arguments apply only to the pre-smoother ('PRE') + or to the post-smoother ('POST'). If pos is not present, + the other arguments are applied to both smoothers. + If the preconditioner is one-level or the parameter identified by what + does not concern the smoothers, pos is ignored.
@@ -96,14 +143,892 @@ as follows:

-call mld_precinit(p,ptype,info) +call mld_precset(p,what,val,info)

+However, in this case the optional arguments ilev, ilmax, and pos +cannot be used. +
+

+A variety of preconditioners can be obtained +by a suitable setting of the preconditioner parameters. These parameters +can be logically divided into four groups, i.e., parameters defining + +

    +
  1. the type of multi-level cycle and how many cycles must be applied; +
    2. +
  2. the aggregation algorithm; +
    3. +
  3. the coarse-space correction at the coarsest level (for multi-level + preconditioners only); +
    4. +
  4. the smoother of the multi-level preconditioners, or the one-level + preconditioner. + +

    +

    5. +
+A list of the parameters that can be set, along with their allowed and +default values, is given in Tables 2-8. +For a description of the meaning of the parameters, please +refer also to Section 4. +
+

+Remark 2. A smoother is usually obtained by combining two objects: +a smoother (mld_smoother_type_) and a local solver (mld_sub_solve_), +as specified in Tables 7-8. +For example, the block-Jacobi smoother using +ILU(0) on the blocks is obtained by combining the block-Jacobi smoother +object with the ILU(0) solver object. Similarly, +the hybrid Gauss-Seidel smoother (see Note in Table 7) +is obtained by combining the block-Jacobi smoother object with a single sweep +of the Gauss-Seidel solver object, while the point-Jacobi smoother is the +result of combining the block-Jacobi smoother object with a single sweep +of the pointwise-Jacobi solver object. However, for simplicity, shortcuts are +provided to set point-Jacobi, hybrid (forward) Gauss-Seidel, and +hybrid backward Gauss-Seidel, i.e., the previous smoothers can be defined +by setting only mld_smoother_type_ to appropriate values (see +Tables 7), i.e., without setting +mld_sub_solve_ too. + +

+The smoother and solver objects are arranged in a +hierarchical manner. When specifying a smoother object, its parameters, +including the local solver, are set to their default values, and when a solver +object is specified, its defaults are also set, overriding in both +cases any previous settings even if explicitly specified. Therefore if +the user sets a smoother, and wishes to use a solver +different from the default one, the call to set the solver must come +after the call to set the smoother. + +

+Similar considerations apply to the point-Jacobi, Gauss-Seidel and block-Jacobi +coarsest-level solvers, and shortcuts are available +in this case too (see Table 5). +
+

+Remark 3. In general, a coarsest-level solver cannot be used with +both the replicated and distributed coarsest-matrix layout; +therefore, setting the solver after the layout may change the layout. +Similarly, setting the layout after the solver may change the solver. + +

+More precisely, UMFPACK and SuperLU require the coarsest-level +matrix to be replicated, while SuperLU_Dist requires it to be distributed. +In these cases, setting the coarsest-level solver implies that +the layout is redefined according to the solver, ovverriding any +previous settings. MUMPS, point-Jacobi, +hybrid Gauss-Seidel and block-Jacobi can be applied to +replicated and distributed matrices, thus their choice +does not modify any previously specified layout. +It is worth noting that, when the matrix is replicated, +the point-Jacobi, hybrid Gauss-Seidel and block-Jacobi solvers +reduce to the corresponding local solver objects (see Remark 2). +For the point-Jacobi and Gauss-Seidel solvers, these objects +correspond to a single point-Jacobi sweep and a single +Gauss-Seidel sweep, respectively, which are very poor solvers. + +

+On the other hand, the distributed layout can be used with any solver +but UMFPACK and SuperLU; therefore, if any of these two solvers has already +been selected, the coarsest-level solver is changed to block-Jacobi, +with the previously chosen solver applied to the local blocks. +Likewise, the replicated layout can be used with any solver but SuperLu_Dist; +therefore, if SuperLu_Dist has been previously set, the coarsest-level +solver is changed to the default sequential solver.

-

+

+
+ + + +
Table 2: +Parameters defining the multi-level cycle and the number of cycles to +be applied. +
+
+ + + + + + + + + + + + + + + + + + + +
whatDATA TYPEvalDEFAULTCOMMENTS
mld_ml_cycle_ +

+ML_CYCLE

character(len=*)'VCYCLE' +

+'WCYCLE' +

+'KCYCLE' +

+'MULT' +

+'ADD'

'VCYCLE'Multi-level cycle: V-cycle, W-cycle, K-cycle, hybrid Multiplicative Schwarz, + and Additive Schwarz. +

+Note that hybrid Multiplicative Schwarz is equivalent to V-cycle and + is included for compatibility with previous versions of MLD2P4.

mld_outer_sweeps_ +

+OUTER_SWEEPS

integerAny integer +

+number $\ge 1$

1Number of multi-level cycles.
+
+
+

+
+ +

+

+
+ + + +
Table 3: +Parameters defining the aggregation algorithm. +
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
whatDATA TYPEvalDEFAULTCOMMENTS
mld_min_coarse_size_ +

+MIN_COARSE_SIZE

integerAny number +

+$> 0$

 +$\lfloor 40 \sqrt[3]{n} \rfloor$, where $n$ is the dimension + of the matrix at the finest levelCoarse size threshold. The aggregation stops + if the global number of variables of the + computed coarsest matrix + is lower than or equal to this threshold + (see Note).
mld_min_cr_ratio_ +

+MIN_CR_RATIO

realAny number +

+$> 1$

1.5Minimum coarsening ratio. The aggregation stops + if the ratio between the matrix dimensions + at two consecutive levels is lower than or equal to this + threshold (see Note).
mld_max_levs_ +

+MAX_LEVS

integerAny integer +

+number $> 1$

20Maximum number of levels. The aggregation stops + if the number of levels reaches this value (see Note).
mld_par_aggr_alg_ +

+PAR_AGGR

character(len=*)'DEC', 'SYMDEC''DEC'Parallel aggregation algorithm. +

+Currently, only the + decoupled aggregation (DEC) is available; the + SYMDEC option applies decoupled + aggregation to the sparsity pattern + of $A+A^T$.

mld_aggr_type_ +

+AGGR_TYPE

character(len=*)'VMB''VMB'Type of aggregation algorithm: currently, the scalar aggregation + algorithm by Vanek, Mandel and Brezina is implemented + [29].
mld_aggr_prol_ +

+AGGR_PROL

character(len=*)'SMOOTHED', 'UNSMOOTHED''SMOOTHED'Prolongator used by the aggregation algorithm: smoothed or unsmoothed + (i.e., tentative prolongator).
Note. The aggregation algorithm stops when +at least one of the following criteria is met: +the coarse size threshold, the
maximum coarsening ratio, or the maximum number +of levels is reached. Therefore, the actual number of levels may be
smaller than the specified maximum number +of levels.
+
+
+

+
+ +

+

+
+ + + +
Table 4: +Parameters defining the aggregation algorithm (continued). +
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
whatDATA TYPEvalDEFAULTCOMMENTS
mld_aggr_ord_ +

+AGGR_ORD

character(len=*)'NATURAL' +

+'DEGREE'

'NATURAL'Initial ordering of indices for the aggregation + algorithm: either natural ordering or sorted by + descending degrees of the nodes in the + matrix graph.
mld_aggr_thresh_ +

+AGGR_THRESH

real(kind_parameter)Any real +

+number $\in [0, 1]$

0.05The threshold $\theta$ in the aggregation algorithm + (see Note).
mld_aggr_omega_alg_ +

+AGGR_OMEGA_ALG

character(len=*)'EIG_EST' +

+'USER_CHOICE'

'EIG_EST'How the damping parameter $\omega$ in the + smoothed aggregation is obtained: + either via an estimate of the spectral radius of + $D^{-1}A$, where $A$ is the matrix at the current + level and $D$ is the diagonal matrix with + the same diagonal entires as $A$, or explicily + specified by the user.
mld_aggr_eig_ +

+AGGR_EIG

character(len=*)'A_NORMI''A_NORMI'How to estimate the spectral radius of $D^{-1}A$. + Currently only the infinity norm estimate + is available.
mld_aggr_omega_val_ +

+AGGR_OMEGA_VAL

real(kind_parameter)Any real +

+number $> 0$

 +$4/(3\rho(D^{-1}A))$Damping parameter $\omega$ in the smoothed aggregation algorithm. + It must be set by the user if + USER_CHOICE was specified for + mld_aggr_omega_alg_, + otherwise it is computed by the library, using the + selected estimate of the spectral radius $\rho(D^{-1}A)$ of + $D^{-1}A$.
mld_aggr_filter_ +

+AGGR_FILTER

character(len=*)'FILTER' +

+'NOFILTER'

'NOFILTER'Matrix used in computing the smoothed + prolongator: filtered or unfiltered.
Note. Different thresholds at different levels, such as +those used in [29, Section 5.1], can be easily set by +invoking the rou-
tine set with +the parameter ilev.
+
+
+

+
+ +

+

+
+ + + +
Table 5: +Parameters defining the coarse-space correction at the coarsest +level.
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
whatDATA TYPEvalDEFAULTCOMMENTS
mld_coarse_mat_ +

+COARSE_MAT

character(len=*)'DIST' +

+'REPL'

'REPL'Coarsest matrix layout: distributed among the processes or + replicated on each of them.
mld_coarse_solve_ +

+COARSE_SOLVE

character(len=*)'MUMPS' +

+'UMF' +

+'SLU' +

+'SLUDIST' +

+'JACOBI' +

+'GS' +

+'BJAC'

See Note.Solver used at the coarsest level: sequential + LU from MUMPS, UMFPACK, or SuperLU + (plus triangular solve); + distributed LU from MUMPS or SuperLU_Dist + (plus triangular solve); + point-Jacobi, hybrid Gauss-Seidel or block-Jacobi. +

+Note that UMF and SLU require the coarsest + matrix to be replicated, SLUDIST, JACOBI, + GS and BJAC require it to be + distributed, MUMPS can be used with either + a replicated or a distributed matrix. When any of the previous + solvers is specified, the matrix layout is set to a default + value + which allows the use + value UMFPACK and SuperLU_Dist + are available only in double precision.

mld_coarse_subsolve_ +

+COARSE_SUBSOLVE

character(len=*)'ILU' +

+'ILUT' +

+'MILU' +

+'MUMPS' +

+'SLU' +

+'UMF'

See Note.Solver for the diagonal blocks of the coarse matrix, + in case the block Jacobi solver + is chosen as coarsest-level solver: ILU($p$), ILU($p,t$), + MILU($p$), LU from MUMPS, SuperLU or UMFPACK + (plus triangular solve). + Note that UMFPACK and SuperLU_Dist + are available only in double precision.
Note. Defaults for mld_coarse_solve_ and +mld_coarse_subsolve_ are chosen in the following order:
single precision version - MUMPS if installed, + then SLU if installed, + ILU otherwise;
double precision version - UMF if installed, + then MUMPS if installed, then SLU if + installed, ILU otherwise.
+
+
+

+
+ +

+

+
+ + + +
Table 6: +Parameters defining the coarse-space correction at the coarsest +level (continued).
+
+ + + + + + + + + + + + + + + + + + + + + + + + + +
whatDATA TYPEvalDEFAULTCOMMENTS
mld_coarse_sweeps_ +

+COARSE_SWEEPS

integerAny integer +

+number $> 0$

10Number of sweeps when JACOBI, GS or BJAC + is chosen as coarsest-level solver.
mld_coarse_fillin_ +

+COARSE_FILLIN

integerAny integer +

+number $\ge 0$

0Fill-in level $p$ of the ILU factorizations.
mld_coarse_iluthrs_ +

+COARSE_ILUTHRS

real(kind_parameter)Any real +

+number $\ge 0$

0Drop tolerance $t$ in the ILU($p,t$) factorization.
+
+
+

+
+ +

+

+
+ + + +
Table 7: +Parameters defining the smoother or the details of the one-level preconditioner. +
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+what DATA TYPE val DEFAULT +COMMENTS
mld_smoother_type_ +

+SMOOTHER_TYPE

character(len=*) + 'JACOBI' +

+'GS' +

+'BGS' +

+'BJAC' + +

+'AS' +

'FBGS' + Type of smoother used in the multi-level preconditioner: + point-Jacobi, hybrid (forward) Gauss-Seidel, + hybrid backward Gauss-Seidel, block-Jacobi, and + Additive Schwarz. +

+It is ignored by one-level preconditioners.

+mld_sub_solve_ +

+SUB_SOLVE

character(len=*) + 'JACOBI' +

+'GS' +

+'BGS' +

+'ILU' +

+'ILUT' +

+'MILU' +

+'MUMPS' +

+'SLU' +

+'UMF' +

GS and BGS for pre- and post-smoothers + of multi-level preconditioners, respectively +

+ILU for block-Jacobi and Additive Schwarz + one-level preconditioners +

The local solver to be used with the smoother or one-level + preconditioner (see Remark 2, page 24): point-Jacobi, + hybrid (forward) Gauss-Seidel, hybrid backward + Gauss-Seidel, ILU($p$), ILU($p,t$), MILU($p$), + LU from MUMPS, SuperLU or UMFPACK + (plus triangular solve). See Note for details on hybrid + Gauss-Seidel.
+mld_moother_sweeps_ +

+SMOOTHER_SWEEPS

integer + Any integer +

+number $\ge 0$ +

1 + Number of sweeps of the smoother or one-level preconditioner. + In the multi-level case, no pre-smother or + post-smoother is used if this parameter is set to 0 + together with pos='PRE' or pos='POST, + respectively.
+mld_sub_ovr_ +

+SUB_OVR

integer + Any integer +

+number $\ge 0$ +

1 + Number of overlap layers, for Additive Schwarz only.
+
+

+
+ +

+

+
+ + + +
Table 8: +Parameters defining the smoother or the details of the one-level preconditioner +(continued).
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+what DATA TYPE val DEFAULT +COMMENTS
+mld_sub_restr_ +

+SUB_RESTR

character(len=*) + 'HALO' +

+'NONE' +

'HALO' + Type of restriction operator, for Additive Schwarz only: + HALO for taking into account the overlap, NONE + for neglecting it. +

+Note that HALO must be chosen for + the classical Addditive Schwarz smoother and its RAS variant.

+mld_sub_prol_ +

+SUB_PROL

character(len=*) + 'SUM' +

+'NONE' +

'NONE' + Type of prolongation operator, for Additive Schwarz only: + SUM for adding the contributions from the overlap, NONE + for neglecting them. +

+Note that SUM must be chosen for the classical Additive + Schwarz smoother, and NONE for its RAS variant.

+mld_sub_fillin_ +

+SUB_FILLIN

integer + Any integer +

+number $\ge 0$ +

0 + Fill-in level $p$ of the incomplete LU factorizations.
+mld_sub_iluthrs_ +

+SUB_ILUTHRS

real(kind_parameter) + Any real number $\ge 0$ + 0 + Drop tolerance $t$ in the ILU($p,t$) factorization.
+
+

+
+ +

+ +

+

+ + +next + +up + +previous + +contents +
+ Next: Subroutine build + Up: User Interface + Previous: Subroutine init +   Contents + diff --git a/docs/html/node18.html b/docs/html/node18.html index 82755007..6e9e191a 100644 --- a/docs/html/node18.html +++ b/docs/html/node18.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Subroutine set - +Subroutine build + @@ -20,52 +20,52 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + - next - + up - previous - contents
- Next: Subroutine bld - Up: User Interface - Previous: Subroutine init -   Next: Subroutine hierarchy_build + Up: User Interface + Previous: Subroutine set +   Contents

-

+


-Subroutine set +Subroutine build

-call p%set(what,val,info [,ilev, ilmax, pos]) - +call p%build(a,desc_a,info) +

-This routine sets the parameters defining the preconditioner p. More -precisely, the parameter identified by what is assigned the value -contained in val. +This routine builds the one-level preconditioner p according to the requirements +made by the user through the routines init and set +(see Sections 6.4 and 6.5 for multi-level preconditioners).

Arguments @@ -73,67 +73,29 @@ contained in val.

- - + + - + - - + + - + - - - - - - - - - - - - - - - - - - - +
whatinteger, intent(in) or character(len=*).
atype(psb_xspmat_type), intent(in).
 The parameter to be set. It can be specified by - a predefined constant, or through its name; the string - is case-insensitive. See also - Tables 2-8.The sparse matrix structure containing the local part of the + matrix to be preconditioned. Note that x must be chosen according + to the real/complex, single/double precision version of MLD2P4 under use. + See the PSBLAS User's Guide for details [17].
val integer or character(len=*) or - real(psb_spk_) or real(psb_dpk_), - intent(in).
desc_atype(psb_desc_type), intent(in).
 The value of the parameter to be set. The list of allowed - values and the corresponding data types is given in - Tables 2-8. - When the value is of type character(len=*), - it is also treated as case insensitive.The communication descriptor of a. See the PSBLAS User's Guide for + details [17].
info integer, intent(out).
 Error code. If no error, 0 is returned. See Section 8 - for details.
ilevinteger, optional, intent(in).
 For the multi-level preconditioner, the level at which the - preconditioner parameter has to be set. - The levels are numbered in increasing - order starting from the finest one, i.e., level 1 is the finest level. - If ilev is not present, the parameter identified by what - is set at all the appropriate levels (see - Tables 2-8).
ilmaxinteger, optional, intent(in).
 For the multi-level preconditioner, when both - ilev and ilmax are present, the settings - are applied at all levels ilev:ilmax. When - ilev is present but ilmax is not, then - the default is ilmax=ilev. - The levels are numbered in increasing - order starting from the finest one, i.e., level 1 is the finest level.
poscharater(len=*), optional, intent(in).
 Whether the other arguments apply only to the pre-smoother ('PRE') - or to the post-smoother ('POST'). If pos is not present, - the other arguments are applied to both smoothers. - If the preconditioner is one-level or the parameter identified by what - does not concern the smoothers, pos is ignored.Error code. If no error, 0 is returned. See Section 8 for details.
@@ -143,890 +105,38 @@ as follows:

-call mld_precset(p,what,val,info) - -
- -

-However, in this case the optional arguments ilev, ilmax, and pos -cannot be used. -
-

-A variety of preconditioners can be obtained -by a suitable setting of the preconditioner parameters. These parameters -can be logically divided into four groups, i.e., parameters defining - -

    -
  1. the type of multi-level cycle and how many cycles must be applied; -
    2. -
  2. the aggregation algorithm; -
    3. -
  3. the coarse-space correction at the coarsest level (for multi-level - preconditioners only); -
    4. -
  4. the smoother of the multi-level preconditioners, or the one-level - preconditioner. - -

    -

    5. -
-A list of the parameters that can be set, along with their allowed and -default values, is given in Tables 2-8. -For a description of the meaning of the parameters, please -refer also to Section 4. -
-

-Remark 2. A smoother is usually obtained by combining two objects: -a smoother (mld_smoother_type_) and a local solver (mld_sub_solve_), -as specified in Tables 7-8. -For example, the block-Jacobi smoother using -ILU(0) on the blocks is obtained by combining the block-Jacobi smoother -object with the ILU(0) solver object. Similarly, -the hybrid Gauss-Seidel smoother (see Note in Table 7) -is obtained by combining the block-Jacobi smoother object with a single sweep -of the Gauss-Seidel solver object, while the point-Jacobi smoother is the -result of combining the block-Jacobi smoother object with a single sweep -of the pointwise-Jacobi solver object. However, for simplicity, shortcuts are -provided to set point-Jacobi, hybrid (forward) Gauss-Seidel, and -hybrid backward Gauss-Seidel, i.e., the previous smoothers can be defined -by setting only mld_smoother_type_ to appropriate values (see -Tables 7), i.e., without setting -mld_sub_solve_ too. - -

-The smoother and solver objects are arranged in a -hierarchical manner. When specifying a smoother object, its parameters, -including the local solver, are set to their default values, and when a solver -object is specified, its defaults are also set, overriding in both -cases any previous settings even if explicitly specified. Therefore if -the user sets a smoother, and wishes to use a solver -different from the default one, the call to set the solver must come -after the call to set the smoother. - -

-Similar considerations apply to the point-Jacobi, Gauss-Seidel and block-Jacobi -coarsest-level solvers, and shortcuts are available -in this case too (see Table 5). -
-

-Remark 3. In general, a coarsest-level solver cannot be used with -both the replicated and distributed coarsest-matrix layout, and vice versa; -therefore, setting the solver after the layout may change the layout, and setting -the layout after the solver may change the solver, if the choices of the two -parameters do not agree. - -

-More precisely, UMFPACK and SuperLU require the coarsest-level -matrix to be replicated, while SuperLU_Dist requires it to be distributed. -In these cases, setting the coarsest-level solver implies that -the layout is redefined according to the solver, ovverriding any -previous settings. MUMPS, point-Jacobi, -hybrid Gauss-Seidel and block-Jacobi can be applied to -replicated and distributed matrices, thus their choice -does not modify any previously specified layout. -It is worth noting that, when the matrix is replicated, -the point-Jacobi, hybrid Gauss-Seidel and block-Jacobi solvers -reduce to the corresponding local solver objects (see Remark 2). -For the point-Jacobi and Gauss-Seidel solvers, these objects -correspond to a single point-Jacobi sweep and a single -Gauss-Seidel sweep, respectively, which are very poor solvers. - -

-On the other hand, the distributed layout can be used with any solver -but UMFPACK and SuperLU; therefore, if any of these two solvers has already -been selected, the coarsest-level solver is changed to block-Jacobi, -with the previously chosen solver applied to the local blocks. -Likewise, the replicated layout can be used with any solver but SuperLu_Dist; -therefore, if SuperLu_Dist has been previously set, the coarsest-level -solver is changed to the default sequential solver. - -

-

-
- - - -
Table 2: -Parameters defining the multi-level cycle and the number of cycles to -be applied. -
-
- - - - - - - - - - - - - - - - - - - -
whatDATA TYPEvalDEFAULTCOMMENTS
mld_ml_cycle_ -

-ML_CYCLE

character(len=*)'VCYCLE' -

-'WCYCLE' -

-'KCYCLE' -

-'MULT' -

-'ADD'

'VCYCLE'Multi-level cycle: V-cycle, W-cycle, K-cycle, hybrid Multiplicative Schwarz, - and Additive Schwarz. -

-Note that hybrid Multiplicative Schwarz is equivalent to V-cycle and - is included for compatibility with previous versions of MLD2P4.

mld_outer_sweeps_ -

-OUTER_SWEEPS

integerAny integer -

-number $\ge 1$

1Number of multi-level cycles.
-
-
-

-
- -

-

-
- - - -
Table 3: -Parameters defining the aggregation algorithm. -
-
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
whatDATA TYPEvalDEFAULTCOMMENTS
mld_min_coarse_size_ -

-MIN_COARSE_SIZE

integerAny number -

-$> 0$

 -$\lfloor 40 \sqrt[3]{n} \rfloor$, where $n$ is the dimension - of the matrix at the finest levelCoarse size threshold. The aggregation stops - if the global number of variables of the - computed coarsest matrix - is lower than or equal to this threshold - (see Note).
mld_min_cr_ratio_ -

-MIN_CR_RATIO

realAny number -

-$> 1$

1.5Minimum coarsening ratio. The aggregation stops - if the ratio between the matrix dimensions - at two consecutive levels is lower than or equal to this - threshold (see Note).
mld_max_levs_ -

-MAX_LEVS

integerAny integer -

-number $> 1$

20Maximum number of levels. The aggregation stops - if the number of levels reaches this value (see Note).
mld_par_aggr_alg_ -

-PAR_AGGR

character(len=*)'DEC', 'SYMDEC''DEC'Parallel aggregation algorithm. -

-Currently, only the - decoupled aggregation (DEC) is available; the - SYMDEC option applies decoupled - aggregation to the sparsity pattern - of $A+A^T$.

mld_aggr_type_ -

-AGGR_TYPE

character(len=*)'VMB''VMB'Type of aggregation algorithm: currently, the scalar aggregation - algorithm by Vanek, Mandel and Brezina is implemented - [27].
mld_aggr_prol_ -

-AGGR_PROL

character(len=*)'SMOOTHED', 'UNSMOOTHED''SMOOTHED'Prolongator used by the aggregation algorithm: smoothed or unsmoothed - (i.e., tentative prolongator).
Note. The aggregation algorithm stops when -at least one of the following criteria is met: -the coarse size threshold, the
maximum coarsening ratio, or the maximum number -of levels is reached. Therefore, the actual number of levels may be
smaller than the specified maximum number -of levels.
-
-
-

-
- -

-

-
- - - -
Table 4: -Parameters defining the aggregation algorithm (continued). -
-
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
whatDATA TYPEvalDEFAULTCOMMENTS
mld_aggr_ord_ -

-AGGR_ORD

character(len=*)'NATURAL' -

-'DEGREE'

'NATURAL'Initial ordering of indices for the aggregation - algorithm: either natural ordering or sorted by - descending degrees of the nodes in the - matrix graph.
mld_aggr_thresh_ -

-AGGR_THRESH

real(kind_parameter)Any real -

-number $\in [0, 1]$

0.05The threshold $\theta$ in the aggregation algorithm - (see Note).
mld_aggr_omega_alg_ -

-AGGR_OMEGA_ALG

character(len=*)'EIG_EST' -

-'USER_CHOICE'

'EIG_EST'How the damping parameter $\omega$ in the - smoothed aggregation is obtained: - either via an estimate of the spectral radius of - $D^{-1}A$, or explicily - specified by the user.
mld_aggr_eig_ -

-AGGR_EIG

character(len=*)'A_NORMI''A_NORMI'How to estimate the spectral radius of $D^{-1}A$. - Currently only the infinity norm estimate - is available.
mld_aggr_omega_val_ -

-AGGR_OMEGA_VAL

real(kind_parameter)Any real -

-number $> 0$

 -$4/(3\rho(D^{-1}A))$Damping parameter $\omega$ in the smoothed aggregation algorithm. - It must be set by the user if - USER_CHOICE was specified for - mld_aggr_omega_alg_, - otherwise it is computed by the library, using the - selected estimate of the spectral radius $\rho(D^{-1}A)$ of - $D^{-1}A$.
mld_aggr_filter_ -

-AGGR_FILTER

character(len=*)'FILTER' -

-'NOFILTER'

'NOFILTER'Matrix used in computing the smoothed - prolongator: filtered or unfiltered.
Note. Different thresholds at different levels, such as -those used in [27, Section 5.1], can be easily set by -invoking the rou-
tine set with -the parameter ilev.
-
-
-

-
- -

-

-
- - - -
Table 5: -Parameters defining the coarse-space correction at the coarsest -level.
-
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
whatDATA TYPEvalDEFAULTCOMMENTS
mld_coarse_mat_ -

-COARSE_MAT

character(len=*)'DIST' -

-'REPL'

'REPL'Coarsest matrix layout: distributed among the processes or - replicated on each of them.
mld_coarse_solve_ -

-COARSE_SOLVE

character(len=*)'MUMPS' -

-'UMF' -

-'SLU' -

-'SLUDIST' -

-'JACOBI' -

-'GS' -

-'BJAC'

See Note 1Solver used at the coarsest level: sequential - LU from MUMPS, UMFPACK, or SuperLU - (plus triangular solve); - distributed LU from MUMPS or SuperLU_Dist - (plus triangular solve); - point-Jacobi, hybrid Gauss-Seidel (see Note 2) or block-Jacobi. -

-Note that UMF and SLU require the coarsest - matrix to be replicated, SLUDIST, JACOBI, - GS and BJAC require it to be - distributed, MUMPS can be used with either - a replicated or a distributed matrix. When any of the previous - solvers is specified, the matrix layout is set to a default - value - which allows the use - value UMFPACK and SuperLU_Dist - are available only in double precision.

mld_coarse_subsolve_ -

-COARSE_SUBSOLVE

character(len=*)'ILU' -

-'ILUT' -

-'MILU' -

-'MUMPS' -

-'SLU' -

-'UMF'

See Note 1Solver for the diagonal blocks of the coarse matrix, - in case the block Jacobi solver - is chosen as coarsest-level solver: ILU($p$), ILU($p,t$), - MILU($p$), LU from MUMPS, SuperLU or UMFPACK - (plus triangular solve). - Note that UMFPACK and SuperLU_Dist - are available only in double precision.
Note 1. Defaults for mld_coarse_solve_ and -mld_coarse_subsolve_ are chosen in the following order:
single precision version - MUMPS if installed, - then SLU if installed, - ILU otherwise;
double precision version - UMF if installed, - then MUMPS if installed, then SLU if - installed, ILU otherwise.
Note 2. The hybrid Gauss-Seidel method is -between the Gauss-Seidel and Jacobi methods: at each iteration, the process-
es use the most recent values of their own local variables, and the values of -the non-local variables computed at the previ-
ous iteration.
-
-
-

-
+call mld_precbld(p,what,val,info) -

-

-
- - - -
Table 6: -Parameters defining the coarse-space correction at the coarsest -level (continued).
-
- - - - - - - - - - - - - - - - - - - - - - - - - -
whatDATA TYPEvalDEFAULTCOMMENTS
mld_coarse_sweeps_ -

-COARSE_SWEEPS

integerAny integer -

-number $> 0$

10Number of sweeps when JACOBI, GS or BJAC - is chosen as coarsest-level solver.
mld_coarse_fillin_ -

-COARSE_FILLIN

integerAny integer -

-number $\ge 0$

0Fill-in level $p$ of the ILU factorizations.
mld_coarse_iluthrs_ -

-COARSE_ILUTHRS

real(kind_parameter)Any real -

-number $\ge 0$

0Drop tolerance $t$ in the ILU($p,t$) factorization.
-
-

-

-

-
- - - -
Table 7: -Parameters defining the smoother or the details of the one-level preconditioner. -
-
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-what DATA TYPE val DEFAULT -COMMENTS
mld_smoother_type_ -

-SMOOTHER_TYPE

character(len=*) - 'JACOBI' -

-'GS' -

-'BGS' -

-'BJAC' - -

-'AS' -

'FBGS' - Type of smoother used in the multi-level preconditioner: - point-Jacobi, hybrid (forward) Gauss-Seidel, - hybrid backward Gauss-Seidel, block-Jacobi, and - Additive Schwarz. See Note for details on hybrix Gauss-Seidel. -

-It is ignored by one-level preconditioners.

-mld_sub_solve_ -

-SUB_SOLVE

character(len=*) - 'JACOBI' -

-'GS' -

-'BGS' -

-'ILU' -

-'ILUT' -

-'MILU' -

-'MUMPS' -

-'SLU' -

-'UMF' -

GS and BGS for pre- and post-smoothers - of multi-level preconditioners, respectively -

-ILU for block-Jacobi and Additive Schwarz - one-level preconditioners -

The local solver to be used with the smoother or one-level - preconditioner (see Remark 2, page 24): point-Jacobi, - hybrid (forward) Gauss-Seidel, hybrid backward - Gauss-Seidel, ILU($p$), ILU($p,t$), MILU($p$), - LU from MUMPS, SuperLU or UMFPACK - (plus triangular solve). See Note for details on hybrid - Gauss-Seidel.
-mld_moother_sweeps_ -

-SMOOTHER_SWEEPS

integer - Any integer -

-number $\ge 0$ -

1 - Number of sweeps of the smoother or one-level preconditioner. - In the multi-level case, no pre-smother or - post-smoother is used if this parameter is set to 0 - together with pos='PRE' or pos='POST, - respectively.
-mld_sub_ovr_ -

-SUB_OVR

integer - Any integer -

-number $\ge 0$ -

1 - Number of overlap layers, for Additive Schwarz only.
-Note. The hybrid Gauss-Seidel method is -between the Gauss-Seidel and Jacobi methods: at each iteration, the processes use the
-most recent values of their own local variables, and the values of -the non-local variables computed at the previous iteration.
-
-

-
- -

-

-
- - - -
Table 8: -Parameters defining the smoother or the details of the one-level preconditioner -(continued).
-
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-what DATA TYPE val DEFAULT -COMMENTS
-mld_sub_restr_ -

-SUB_RESTR

character(len=*) - 'HALO' -

-'NONE' -

'HALO' - Type of restriction operator, for Additive Schwarz only: - HALO for taking into account the overlap, NONE - for neglecting it.
-mld_sub_prol_ -

-SUB_PROL

character(len=*) - 'SUM' -

-'NONE' -

'NONE' - Type of prolongation operator, for Additive Schwarz only: - SUM for adding the contributions from the overlap, NONE - for neglecting them.
-mld_sub_fillin_ -

-SUB_FILLIN

integer - Any integer -

-number $\ge 0$ -

0 - Fill-in level $p$ of the incomplete LU factorizations.
-mld_sub_iluthrs_ -

-SUB_ILUTHRS

real(kind_parameter) - Any real number $\ge 0$ - 0 - Drop tolerance $t$ in the ILU($p,t$) factorization.
-
-

-
+In this case, the routine can be used to build multi-level preconditioners too.

- next - + up - previous - contents
- Next: Subroutine bld - Up: User Interface - Previous: Subroutine init -   Next: Subroutine hierarchy_build + Up: User Interface + Previous: Subroutine set +   Contents diff --git a/docs/html/node19.html b/docs/html/node19.html index 40c6ae2c..35bd97df 100644 --- a/docs/html/node19.html +++ b/docs/html/node19.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Subroutine bld - +Subroutine hierarchy_build + @@ -20,52 +20,52 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + - next - + up - previous - contents
- Next: Subroutine hierarchy_bld - Up: User Interface - Previous: Subroutine set -   Next: Subroutine smoothers_build + Up: User Interface + Previous: Subroutine build +   Contents

-

+


-Subroutine bld +Subroutine hierarchy_build

-call p%bld(a,desc_a,info) +call p%hierarchy_build(a,desc_a,info)

-This routine builds the one-level preconditioner p according to the requirements -made by the user through the routines init and set -(see Sections 6.4 and 6.5 for multi-level preconditioners). +This routine builds the hierarchy of matrices and restriction/prolongation +operators for the multi-level preconditioner p, according to the requirements +made by the user through the routines init and set.

Arguments @@ -79,9 +79,10 @@ made by the user through the routines init and set

  The sparse matrix structure containing the local part of the matrix to be preconditioned. Note that x must be chosen according - to the real/complex, single/double precision version of MLD2P4 under use. + to the real/complex, +single/double precision version of MLD2P4 under use. See the PSBLAS User's Guide for details [16].
desc_a type(psb_desc_type), intent(in).
  The communication descriptor of a. See the PSBLAS User's Guide for details [16].
info integer, intent(out).
 Error code. If no error, 0 is returned. See Section 8 for details.Error code. If no error, 0 is returned. See Section 8 for details.

-For compatibility with the previous versions of MLD2P4, this routine can be also invoked -as follows: - -

-

-call mld_precbld(p,what,val,info) - -
- -

-In this case, the routine can be used to build multi-level preconditioners too.

- -

-

- - -next - -up - -previous - -contents -
- Next: Subroutine hierarchy_bld - Up: User Interface - Previous: Subroutine set -   Contents - +
diff --git a/docs/html/node2.html b/docs/html/node2.html index 82730de5..a560c26e 100644 --- a/docs/html/node2.html +++ b/docs/html/node2.html @@ -26,21 +26,21 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous
- Next: Next: General Overview - Up: Up: userhtml - Previous: Previous: Abstract

@@ -53,70 +53,68 @@ Contents -
+ HREF="node26.html">License
  • Adding smoothers and solvers to MLD2P4 -
  • Error Handling -
  • License -
  • Bibliography + HREF="node27.html">Bibliography diff --git a/docs/html/node20.html b/docs/html/node20.html index 08545757..2c381342 100644 --- a/docs/html/node20.html +++ b/docs/html/node20.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Subroutine hierarchy_bld - +Subroutine smoothers_build + @@ -20,52 +20,54 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + - next - + up - previous - contents
    - Next: Subroutine smoothers_bld - Up: User Interface - Previous: Subroutine bld -   Next: Subroutine apply + Up: User Interface + Previous: Subroutine hierarchy_build +   Contents

    -

    +


    -Subroutine hierarchy_bld +Subroutine smoothers_build

    -call p%hierarchy_bld(a,desc_a,info) +call p%smoothers_build(a,desc_a,p,info)

    -This routine builds the hierarchy of matrices and restriction/prolongation -operators for the multi-level preconditioner p, according to the requirements -made by the user through the routines init and set. +This routine builds the smoothers and the coarsest-level solvers for the +multi-level preconditioner p, according to the requirements made by +the user through the routines init and set, and based on the aggregation +hierarchy produced by a previous call to hierarchy_build +(see Section 6.4).

    Arguments @@ -79,10 +81,9 @@ made by the user through the routines init and set.   The sparse matrix structure containing the local part of the matrix to be preconditioned. Note that x must be chosen according - to the real/complex, -single/double precision version of MLD2P4 under use. + to the real/complex, single/double precision version of MLD2P4 under use. See the PSBLAS User's Guide for details [16]. + HREF="node27.html#PSBLASGUIDE">17]. desc_a type(psb_desc_type), intent(in). @@ -90,19 +91,18 @@ single/double precision version of MLD2P4 under use.   The communication descriptor of a. See the PSBLAS User's Guide for details [16]. + HREF="node27.html#PSBLASGUIDE">17]. info integer, intent(out).   -Error code. If no error, 0 is returned. See Section 8 for details. +Error code. If no error, 0 is returned. See Section 8 for details.

    -


    diff --git a/docs/html/node21.html b/docs/html/node21.html index ef097197..65974c7c 100644 --- a/docs/html/node21.html +++ b/docs/html/node21.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Subroutine smoothers_bld - +Subroutine apply + @@ -20,54 +20,68 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + - next - + up - previous - contents
    - Next: Subroutine apply - Up: User Interface - Previous: Subroutine hierarchy_bld -   Next: Subroutine free + Up: User Interface + Previous: Subroutine smoothers_build +   Contents

    -

    +


    -Subroutine smoothers_bld +Subroutine apply

    -call p%smoothers_bld(a,desc_a,p,info) +call p%apply(x,y,desc_a,info [,trans,work])

    -This routine builds the smoothers and the coarsest-level solvers for the -multi-level preconditioner p, according to the requirements made by -the user through the routines init and set, and based on the aggregation -hierarchy produced by a previous call to hierarchy_bld -(see Section 6.4). +This routine computes +$y = op(M^{-1}) x$, where $M$ is a previously built +preconditioner, stored into p, and $op$ +denotes the preconditioner itself or its transpose, according to +the value of trans. +Note that, when MLD2P4 is used with a Krylov solver from PSBLAS, +p%apply is called within the PSBLAS routine psb_krylov +and hence it is completely transparent to the user.

    Arguments @@ -75,35 +89,122 @@ hierarchy produced by a previous call to hierarchy_bld

    - - + + + + + + + + - + - + - + + + + + + + + + + + + +
    atype(psb_xspmat_type), intent(in).
    xtype(kind_parameter), dimension(:), intent(in).
     The local part of the vector $x$. Note that type and + kind_parameter must be chosen according + to the real/complex, single/double precision version of MLD2P4 under use.
    ytype(kind_parameter), dimension(:), intent(out).
     The sparse matrix structure containing the local part of the - matrix to be preconditioned. Note that x must be chosen according - to the real/complex, single/double precision version of MLD2P4 under use. - See the PSBLAS User's Guide for details [16].The local part of the vector $y$. Note that type and + kind_parameter must be chosen according + to the real/complex, single/double precision version of MLD2P4 under use.
    desc_a type(psb_desc_type), intent(in).
     The communication descriptor of a. See the PSBLAS User's Guide for - details [16].The communication descriptor associated to the matrix to be + preconditioned.
    info integer, intent(out).
     Error code. If no error, 0 is returned. See Section 8 for details.Error code. If no error, 0 is returned. See Section 8 for details.
    transcharacter(len=1), optional, intent(in).
     If trans = 'N','n' then +$op(M^{-1}) = M^{-1}$; + if trans = 'T','t' then +$op(M^{-1}) = M^{-T}$ + (transpose of $M^{-1})$; if trans = 'C','c' then +$op(M^{-1}) = M^{-C}$ + (conjugate transpose of $M^{-1})$.
    worktype(kind_parameter), dimension(:), optional, target.
     Workspace. Its size should be at + least 4 * psb_cd_get_local_ cols(desc_a) (see the PSBLAS User's Guide). + Note that type and kind_parameter must be chosen according + to the real/complex, single/double precision version of MLD2P4 under use.

    +For compatibility with the previous versions of MLD2P4, this routine can be also invoked +as follows: + +

    +

    +call mld_precaply(p,what,val,info) + +
    -
    +

    + +

    +

    + + +next + +up + +previous + +contents +
    + Next: Subroutine free + Up: User Interface + Previous: Subroutine smoothers_build +   Contents + diff --git a/docs/html/node22.html b/docs/html/node22.html index dc76cdab..ff93d7a4 100644 --- a/docs/html/node22.html +++ b/docs/html/node22.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Subroutine apply - +Subroutine free + @@ -20,68 +20,50 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + - next - + up - previous - contents
    - Next: Subroutine free - Up: User Interface - Previous: Subroutine smoothers_bld -   Next: Subroutine descr + Up: User Interface + Previous: Subroutine apply +   Contents

    -

    +


    -Subroutine apply +Subroutine free

    -call p%apply(x,y,desc_a,info [,trans,work]) +call p%free(p,info)

    -This routine computes -$y = op(M^{-1})\, x$, where $M$ is a previously built -preconditioner, stored into p, and $op$ -denotes the preconditioner itself or its transpose, according to -the value of trans. -Note that, when MLD2P4 is used with a Krylov solver from PSBLAS, -p%apply is called within the PSBLAS routine psb_krylov -and hence it is completely transparent to the user. +This routine deallocates the preconditioner data structure p.

    Arguments @@ -89,82 +71,11 @@ and hence it is completely transparent to the user.

    - - - - - - - - - - - - - - - - - - - + - - - - - - - - - - - - - +
    xtype(kind_parameter), dimension(:), intent(in).
     The local part of the vector $x$. Note that type and - kind_parameter must be chosen according - to the real/complex, single/double precision version of MLD2P4 under use.
    ytype(kind_parameter), dimension(:), intent(out).
     The local part of the vector $y$. Note that type and - kind_parameter must be chosen according - to the real/complex, single/double precision version of MLD2P4 under use.
    desc_atype(psb_desc_type), intent(in).
     The communication descriptor associated to the matrix to be - preconditioned.
    infointeger, intent(out).integer, intent(out).
     Error code. If no error, 0 is returned. See Section 8 for details.
    transcharacter(len=1), optional, intent(in).
     If trans = 'N','n' then -$op(M^{-1}) = M^{-1}$; - if trans = 'T','t' then -$op(M^{-1}) = M^{-T}$ - (transpose of $M^{-1})$; if trans = 'C','c' then -$op(M^{-1}) = M^{-C}$ - (conjugate transpose of $M^{-1})$.
    worktype(kind_parameter), dimension(:), optional, target.
     Workspace. Its size should be at - least 4 * psb_cd_get_local_ cols(desc_a) (see the PSBLAS User's Guide). - Note that type and kind_parameter must be chosen according - to the real/complex, single/double precision version of MLD2P4 under use.Error code. If no error, 0 is returned. See Section 8 for details.
    @@ -174,37 +85,14 @@ as follows:

    -call mld_precaply(p,what,val,info) +call mld_precfree(p,info)

    -

    - - -next - -up - -previous - -contents -
    - Next: Subroutine free - Up: User Interface - Previous: Subroutine smoothers_bld -   Contents - +
    diff --git a/docs/html/node23.html b/docs/html/node23.html index 79a8e176..ef67731a 100644 --- a/docs/html/node23.html +++ b/docs/html/node23.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Subroutine free - +Subroutine descr + @@ -18,52 +18,53 @@ original version by: Nikos Drakos, CBLU, University of Leeds - - + - next - -up +up + previous - contents
    - Next: Subroutine descr - Up: User Interface - Previous: Subroutine apply -   Next: Adding smoothers and solvers + Up: User Interface + Previous: Subroutine free +   Contents

    -

    +


    -Subroutine free +Subroutine descr

    -call p%free(p,info) +call p%descr(info, [iout])

    -This routine deallocates the preconditioner data structure p. +This routine prints a description of the preconditioner p to the standard output or +to a file. It must be called after hierachy_build and smoothers_build, +or build, have been called.

    Arguments @@ -72,10 +73,17 @@ This routine deallocates the preconditioner data structure p.

    - + + + + + + + - +
    infointeger, intent(out).integer, intent(out).
     Error code. If no error, 0 is returned. See Section 8 for details.
    ioutinteger, intent(in), optional.
     Error code. If no error, 0 is returned. See Section 8 for details.The id of the file where the preconditioner description + will be printed; the default is the standard output.
    @@ -85,7 +93,7 @@ as follows:

    -call mld_precfree(p,info) +call mld_precdescr(p,info [,iout])
    diff --git a/docs/html/node24.html b/docs/html/node24.html index 2db3650a..177340fe 100644 --- a/docs/html/node24.html +++ b/docs/html/node24.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Subroutine descr - +Adding smoothers and solvers to MLD2P4 + @@ -18,89 +18,119 @@ original version by: Nikos Drakos, CBLU, University of Leeds - - + + + - next - -up +up + previous - contents
    - Next: Adding smoothers and solvers - Up: User Interface - Previous: Subroutine free -   Next: Error Handling + Up: userhtml + Previous: Subroutine descr +   Contents

    -

    +


    -Subroutine descr -

    +Adding smoothers and solvers to MLD2P4 +

    -

    -call p%descr(info, [iout]) +Da ampliare e completare - SALVATORE.
    -
    -

    -This routine prints a description of the preconditioner p to the standard output or -to a file. It must be called after hierachy_bld and smoothers_bld, -or bld, have been called. +Completely new smoother and/or solver classes derived from the +base objects in the library may be used without recompiling the +library itself. Once the new smoother/solver class has been +developed, the user can declare a variable of that new type in the +application, and pass that variable to the p%set(solver,info) +call; the new solver object is then dynamically included in the +preconditioner structure.

    -Arguments -

    + +
    +

    +If the user has developed a new type of smoother and/or +solver by extending one of the base MLD2P4 types, and has declared a +variable of the new type in the main program, it is possible to pass +the new smoother/solver variable to the setup routine as follows: +

    +call p%set(smoother,info [,ilev, ilmax,pos]) +
    call p%set(solver,info [,ilev, ilmax,pos]) + +
    +In this way, the variable will act as a mold to which the +preconditioner will conform, even though the MLD2P4 library is not +modified, and thus has no direct knowledge about the new type. +

    + +

    - - + + - + - - + + - +
    infointeger, intent(out).
    smootherclass(mld_x_base_smoother_type)
     Error code. If no error, 0 is returned. See Section 8 for details.The user-defined new smoother to be employed in the + preconditioner.
    ioutinteger, intent(in), optional.
    solverclass(mld_x_base_solver_type)
     The id of the file where the preconditioner description - will be printed; the default is the standard output.The user-defined new solver to be employed in the + preconditioner.

    -For compatibility with the previous versions of MLD2P4, this routine can be also invoked -as follows: - -

    -

    -call mld_precdescr(p,info [,iout]) - -
    - -

    - -

    -

    +
    + + +next + +up + +previous + +contents +
    + Next: Error Handling + Up: userhtml + Previous: Subroutine descr +   Contents + diff --git a/docs/html/node25.html b/docs/html/node25.html index 24f922b0..31ff396e 100644 --- a/docs/html/node25.html +++ b/docs/html/node25.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Adding smoothers and solvers to MLD2P4 - +Error Handling + @@ -19,118 +19,60 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + - next - up - previous - contents
    - Next: Error Handling - Up: Next: License + Up: userhtml - Previous: Subroutine descr -   Previous: Adding smoothers and solvers +   Contents

    -

    +


    -Adding smoothers and solvers to MLD2P4 +Error Handling

    -Da ampliare e completare - SALVATORE. -
    -

    -Completely new smoother and/or solver classes derived from the -base objects in the library may be used without recompiling the -library itself. Once the new smoother/solver class has been -developed, the user can declare a variable of that new type in the -application, and pass that variable to the p%set(solver,info) -call; the new solver object is then dynamically included in the -preconditioner structure. +The error handling in MLD2P4 is based on the PSBLAS (version 2) error +handling. Error conditions are signaled via an integer argument +info; whenever an error condition is detected, an error trace +stack is built by the library up to the top-level, user-callable +routine. This routine will then decide, according to the user +preferences, whether the error should be handled by terminating the +program or by returning the error condition to the user code, which +will then take action, and whether +an error message should be printed. These options may be set by using +the PSBLAS error handling routines; for further details see the PSBLAS +User's Guide [17].

    - -
    -

    -If the user has developed a new type of smoother and/or -solver by extending one of the base MLD2P4 types, and has declared a -variable of the new type in the main program, it is possible to pass -the new smoother/solver variable to the setup routine as follows: -

    -call p%set(smoother,info [,ilev, ilmax,pos]) -
    call p%set(solver,info [,ilev, ilmax,pos]) - -
    -In this way, the variable will act as a mold to which the -preconditioner will conform, even though the MLD2P4 library is not -modified, and thus has no direct knowledge about the new type. -

    - -
    -

    - - - - - - - - - - - - - -
    smootherclass(mld_x_base_smoother_type)
     The user-defined new smoother to be employed in the - preconditioner.
    solverclass(mld_x_base_solver_type)
     The user-defined new solver to be employed in the - preconditioner.
    - -

    -

    - - -next - -up - -previous - -contents -
    - Next: Error Handling - Up: userhtml - Previous: Subroutine descr -   Contents - +
    diff --git a/docs/html/node26.html b/docs/html/node26.html index bb5c875f..0730f26e 100644 --- a/docs/html/node26.html +++ b/docs/html/node26.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Error Handling - +License + @@ -26,51 +26,77 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
    - Next: License - Up: Next: Bibliography + Up: userhtml - Previous: Adding smoothers and solvers -   Previous: Error Handling +   Contents

    -

    +


    -Error Handling +License

    -The error handling in MLD2P4 is based on the PSBLAS (version 2) error -handling. Error conditions are signaled via an integer argument -info; whenever an error condition is detected, an error trace -stack is built by the library up to the top-level, user-callable -routine. This routine will then decide, according to the user -preferences, whether the error should be handled by terminating the -program or by returning the error condition to the user code, which -will then take action, and whether -an error message should be printed. These options may be set by using -the PSBLAS error handling routines; for further details see the PSBLAS -User's Guide [16]. +The MLD2P4 is freely distributable under the following copyright +terms: 

     
 
-
    
+ 
+                           MLD2P4  version 2.1
+  MultiLevel Domain Decomposition Parallel Preconditioners Package
+             based on PSBLAS (Parallel Sparse BLAS version 3.4)
+  
+  (C) Copyright 2008, 2010, 2012, 2017
+
+  Salvatore Filippone    Cranfield University, Cranfield, UK
+  Ambra Abdullahi Hassan University of Rome Tor Vergata, Rome, IT
+  Alfredo Buttari        CNRS-IRIT, Toulouse, FR
+  Pasqua D'Ambra         IAC-CNR, Naples, IT
+  Daniela di Serafino    University of Campania L. Vanvitelli, Caserta, IT
+
+  Redistribution and use in source and binary forms, with or without
+  modification, are permitted provided that the following conditions
+  are met:
+    1. Redistributions of source code must retain the above copyright
+       notice, this list of conditions and the following disclaimer.
+    2. Redistributions in binary form must reproduce the above copyright
+       notice, this list of conditions, and the following disclaimer in the
+       documentation and/or other materials provided with the distribution.
+    3. The name of the MLD2P4 group or the names of its contributors may
+       not be used to endorse or promote products derived from this
+       software without specific written permission.
+ 
+  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+  ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
+  TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
+  PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE MLD2P4 GROUP OR ITS CONTRIBUTORS
+  BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+  CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+  SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+  INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+  CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
+  ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
+  POSSIBILITY OF SUCH DAMAGE.
+

    diff --git a/docs/html/node27.html b/docs/html/node27.html index 189e0679..34989e95 100644 --- a/docs/html/node27.html +++ b/docs/html/node27.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -License - +Bibliography + @@ -26,79 +26,167 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
    - Next: Bibliography - Up: Next: About this document ... + Up: userhtml - Previous: Error Handling -   Previous: License +   Contents -
    -
    +

    - -

    -
    -License -

    - -

    -The MLD2P4 is freely distributable under the following copyright -terms: 

     
-
  
-                           MLD2P4  version 2.1
-  MultiLevel Domain Decomposition Parallel Preconditioners Package
-             based on PSBLAS (Parallel Sparse BLAS version 3.4)
-  
-  (C) Copyright 2008, 2010, 2012, 2017
-
-                      Salvatore Filippone    Cranfield University
-   		      Ambra Abdullahi Hassan University of Rome Tor Vergata
-                      Alfredo Buttari        CNRS-IRIT, Toulouse
-                      Pasqua D'Ambra         ICAR-CNR, Naples
-                      Daniela di Serafino    Second University of Naples
+
    
+Bibliography
+
    
 
- 
-  Redistribution and use in source and binary forms, with or without
-  modification, are permitted provided that the following conditions
-  are met:
-    1. Redistributions of source code must retain the above copyright
-       notice, this list of conditions and the following disclaimer.
-    2. Redistributions in binary form must reproduce the above copyright
-       notice, this list of conditions, and the following disclaimer in the
-       documentation and/or other materials provided with the distribution.
-    3. The name of the MLD2P4 group or the names of its contributors may
-       not be used to endorse or promote products derived from this
-       software without specific written permission.
- 
-  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
-  ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
-  TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
-  PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE MLD2P4 GROUP OR ITS CONTRIBUTORS
-  BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
-  CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
-  SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
-  INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
-  CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
-  ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
-  POSSIBILITY OF SUCH DAMAGE.
-
    +

    +

    1 +
    +M. Brezina, P. Vanek, +A Black-Box Iterative Solver Based on a Two-Level Schwarz Method, +Computing, 63, 1999, 233-263.

    2 +
    +W. L. Briggs, V. E. Henson, S. F.  McCormick, +A Multigrid Tutorial, Second Edition, +SIAM, 2000.

    3 +
    +A. Buttari, P. D'Ambra, D. di Serafino, S. Filippone, +Extending PSBLAS to Build Parallel Schwarz Preconditioners, +in J. Dongarra, K. Madsen, J. Wasniewski, editors, +Proceedings of PARA 04 Workshop on State of the Art +in Scientific Computing, Lecture Notes in Computer Science, +Springer, 2005, 593-602.

    4 +
    +A. Buttari, P. D'Ambra, D. di Serafino, S. Filippone, +2LEV-D2P4: a package of high-performance preconditioners +for scientific and engineering applications, +Applicable Algebra in Engineering, Communications and Computing, +18 (3) 2007, 223-239.

    5 +
    P. D'Ambra, S. Filippone, D. di Serafino, +On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners, +Applied Numerical Mathematics, Elsevier Science, +57 (11-12), 2007, 1181-1196.

    6 +
    +X. C. Cai, M. Sarkis, +A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems, +SIAM Journal on Scientific Computing, 21 (2), 1999, 792-797.

    7 +
    +X. C. Cai, O. B. Widlund, +Domain Decomposition Algorithms for Indefinite Elliptic Problems, +SIAM Journal on Scientific and Statistical Computing, 13 (1), 1992, 243-258.

    8 +
    +T. Chan and T. Mathew, +Domain Decomposition Algorithms, +in A. Iserles, editor, Acta Numerica 1994, 61-143. +Cambridge University Press.

    9 +
    +P. D'Ambra, D. di Serafino, S. Filippone, +MLD2P4: a Package of Parallel Multilevel +Algebraic Domain Decomposition Preconditioners +in Fortran 95, ACM Trans. Math. Softw., 37(3), 2010, art. 30.

    10 +
    +T.A. Davis, +Algorithm 832: UMFPACK - an Unsymmetric-pattern Multifrontal +Method with a Column Pre-ordering Strategy, +ACM Transactions on Mathematical Software, 30, 2004, 196-199. +(See also http://www.cise.ufl.edu/ davis/)

    11 +
    +P.R. Amestoy, C. Ashcraft, O. Boiteau, A. Buttari, J. L'Excellent, C. Weisbecker +Improving multifrontal methods by means of block low-rank representations, +SIAM Journal on Scientific Computing, volume 37 (3), 2015, A1452-A1474. +See also http://mumps.enseeiht.fr.

    12 +
    +J.W. Demmel, S.C. Eisenstat, J.R. Gilbert, X.S. Li and J.W.H. Liu, +A supernodal approach to sparse partial pivoting, +SIAM Journal on Matrix Analysis and Applications, 20 (3), 1999, 720-755.

    13 +
    +J. J. Dongarra, J. Du Croz, I. S. Duff, S. Hammarling, +A set of Level 3 Basic Linear Algebra Subprograms, +ACM Transactions on Mathematical Software, 16 (1) 1990, 1-17.

    14 +
    +J. J. Dongarra, J. Du Croz, S. Hammarling, R. J. Hanson, +An extended set of FORTRAN Basic Linear Algebra Subprograms, +ACM Transactions on Mathematical Software, 14 (1) 1988, 1-17.

    15 +
    +J. J. Dongarra and R. C. Whaley, +A User's Guide to the BLACS v. 1.1, +Lapack Working Note 94, Tech. Rep. UT-CS-95-281, University of +Tennessee, March 1995 (updated May 1997).

    16 +
    +E. Efstathiou, J. G. Gander, +Why Restricted Additive Schwarz Converges Faster than Additive Schwarz, +BIT Numerical Mathematics, 43 (5), 2003, 945-959.

    17 +
    +S. Filippone, A. Buttari, +PSBLAS-3.0 User's Guide. A Reference Guide for the Parallel Sparse BLAS Library, 2012, +available from http://www.ce.uniroma2.it/psblas/.

    18 +
    +Salvatore Filippone and Alfredo Buttari. +Object-Oriented Techniques for Sparse Matrix Computations in Fortran 2003. +ACM Transactions on on Mathematical Software, 38 (4), 2012, art. 23.

    19 +
    +S. Filippone, M. Colajanni, +PSBLAS: A Library for Parallel Linear Algebra +Computation on Sparse Matrices, +ACM Transactions on Mathematical Software, 26 (4), 2000, 527-550.

    20 +
    +W. Gropp, S. Huss-Lederman, A. Lumsdaine, E. Lusk, B. Nitzberg, W. Saphir, M. Snir, +MPI: The Complete Reference. Volume 2 - The MPI-2 Extensions, +MIT Press, 1998.

    21 +
    +C. L. Lawson, R. J. Hanson, D. Kincaid, F. T. Krogh, +Basic Linear Algebra Subprograms for FORTRAN usage, +ACM Transactions on Mathematical Software, 5 (3), 1979, 308-323.

    22 +
    +X. S. Li, J. W. Demmel, SuperLU_DIST: A Scalable Distributed-memory +Sparse Direct Solver for Unsymmetric Linear Systems, +ACM Transactions on Mathematical Software, 29 (2), 2003, 110-140.

    23 +
    +Y. Notay, P. S. Vassilevski, Recursive Krylov-based multigrid cycles, +Numerical Linear Algebra with Applications, 15 (5), 2008, 473-487.

    24 +
    +Y. Saad, +Iterative methods for sparse linear systems, 2nd edition, SIAM, 2003.

    25 +
    +B. Smith, P. Bjorstad, W. Gropp, +Domain Decomposition: Parallel Multilevel Methods for Elliptic +Partial Differential Equations, +Cambridge University Press, 1996.

    26 +
    +M. Snir, S. Otto, S. Huss-Lederman, D. Walker, J. Dongarra, +MPI: The Complete Reference. Volume 1 - The MPI Core, second edition, +MIT Press, 1998.

    27 +
    +K. Stüben, +An Introduction to Algebraic Multigrid, +in A. Schüller, U. Trottenberg, C. Oosterlee, Multigrid, +Academic Press, 2001.

    28 +
    +R. S. Tuminaro, C. Tong, +Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines, in J. Donnelley, editor, Proceedings of SuperComputing 2000, Dallas, 2000.

    29 +
    +P. Vanek, J. Mandel and M. Brezina, +Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems, +Computing, 56 (3) 1996, 179-196. +

    + +


    diff --git a/docs/html/node28.html b/docs/html/node28.html index 95a6d89a..faaa7903 100644 --- a/docs/html/node28.html +++ b/docs/html/node28.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Bibliography - +About this document ... + @@ -18,186 +18,52 @@ original version by: Nikos Drakos, CBLU, University of Leeds - - - -next - + up - previous - contents
    - Next: About this document ... - Up: Up: userhtml - Previous: License -   Previous: Bibliography +   Contents -

    +
    +
    - -

    -Bibliography -

    +

    +About this document ... +

    +

    +This document was generated using the +LaTeX2HTML translator Version 2012 (1.2)

    -

    1 -
    -M. Brezina, P. Vanek, -A Black-Box Iterative Solver Based on a Two-Level Schwarz Method, -Computing, 63, 1999, 233-263.

    2 -
    -A. Buttari, P. D'Ambra, D. di Serafino, S. Filippone, -Extending PSBLAS to Build Parallel Schwarz Preconditioners, -in , J. Dongarra, K. Madsen, J. Wasniewski, editors, -Proceedings of PARA 04 Workshop on State of the Art -in Scientific Computing, Lecture Notes in Computer Science, -Springer, 2005, 593-602.

    3 -
    -A. Buttari, P. D'Ambra, D. di Serafino, S. Filippone, -2LEV-D2P4: a package of high-performance preconditioners -for scientific and engineering applications, -Applicable Algebra in Engineering, Communications and Computing, -18, 3, 2007, 223-239.

    4 -
    P. D'Ambra, S. Filippone, D. di Serafino, -On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners, -Applied Numerical Mathematics, Elsevier Science, -57, 11-12, 2007, 1181-1196. -

    -

    5 -
    -X. C. Cai, M. Sarkis, -A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems, -SIAM Journal on Scientific Computing, 21, 2, 1999, 792-797.

    6 -
    -X. C. Cai, O. B. Widlund, -Domain Decomposition Algorithms for Indefinite Elliptic Problems, -SIAM Journal on Scientific and Statistical Computing, 13, 1, 1992, 243-258.

    7 -
    -T. Chan and T. Mathew, -Domain Decomposition Algorithms, -in A. Iserles, editor, Acta Numerica 1994, 61-143. -Cambridge University Press.

    8 -
    -P. D'Ambra, D. di Serafino, S. Filippone, -MLD2P4: a Package of Parallel Multilevel -Algebraic Domain Decomposition Preconditioners -in Fortran 95, ACM Trans. Math. Softw., 37(3), 2010.

    9 -
    -T.A. Davis, -Algorithm 832: UMFPACK - an Unsymmetric-pattern Multifrontal -Method with a Column Pre-ordering Strategy, -ACM Transactions on Mathematical Software, 30, 2004, 196-199. -(See also http://www.cise.ufl.edu/ davis/) -

    -

    10 -
    -P.R. Amestoy, C. Ashcraft, O. Boiteau, A. Buttari, J. L'Excellent, C. Weisbecker -Improving multifrontal methods by means of block low-rank representations, -SIAM SISC, volume 37, number 3, pages A1452-A1474. -(See also http://mumps.enseeiht.fr) -

    -

    11 -
    -J.W. Demmel, S.C. Eisenstat, J.R. Gilbert, X.S. Li and J.W.H. Liu, -A supernodal approach to sparse partial pivoting, -SIAM Journal on Matrix Analysis and Applications, 20, 3, 1999, 720-755.

    12 -
    -J. J. Dongarra, J. Du Croz, I. S. Duff, S. Hammarling, -A set of Level 3 Basic Linear Algebra Subprograms, -ACM Transactions on Mathematical Software, 16, 1990, 1-17.

    13 -
    -J. J. Dongarra, J. Du Croz, S. Hammarling, R. J. Hanson, -An extended set of FORTRAN Basic Linear Algebra Subprograms, -ACM Transactions on Mathematical Software, 14, 1988, 1-17.

    14 -
    -J. J. Dongarra and R. C. Whaley, -A User's Guide to the BLACS v. 1.1, -Lapack Working Note 94, Tech. Rep. UT-CS-95-281, University of -Tennessee, March 1995 (updated May 1997).

    15 -
    -E. Efstathiou, J. G. Gander, -Why Restricted Additive Schwarz Converges Faster than Additive Schwarz, -BIT Numerical Mathematics, 43, 2003, 945-959.

    16 -
    -S. Filippone, A. Buttari, -PSBLAS-3.0 User's Guide. A Reference Guide for the Parallel Sparse BLAS Library, 2012, -available from http://www.ce.uniroma2.it/psblas/. - -

    -

    17 -
    -Salvatore Filippone and Alfredo Buttari. -Object-Oriented Techniques for Sparse Matrix Computations in Fortran - 2003. -ACM Trans. on Math Software, 38(4), 2012. - -

    -

    18 -
    -S. Filippone, M. Colajanni, -PSBLAS: A Library for Parallel Linear Algebra -Computation on Sparse Matrices, -ACM Transactions on Mathematical Software, 26, 4, 2000, 527-550.

    19 -
    -W. Gropp, S. Huss-Lederman, A. Lumsdaine, E. Lusk, B. Nitzberg, W. Saphir, M. Snir, -MPI: The Complete Reference. Volume 2 - The MPI-2 Extensions, -MIT Press, 1998.

    20 -
    -C. L. Lawson, R. J. Hanson, D. Kincaid, F. T. Krogh, -Basic Linear Algebra Subprograms for FORTRAN usage, -ACM Transactions on Mathematical Software, 5, 1979, 308-323.

    21 -
    -X. S. Li, J. W. Demmel, SuperLU_DIST: A Scalable Distributed-memory -Sparse Direct Solver for Unsymmetric Linear Systems, -ACM Transactions on Mathematical Software, 29, 2, 2003, 110-140.

    22 -
    -Y. Saad, -Iterative methods for sparse linear systems, 2nd edition, -SIAM, 2003 - -

    -

    23 -
    -B. Smith, P. Bjorstad, W. Gropp, -Domain Decomposition: Parallel Multilevel Methods for Elliptic -Partial Differential Equations, -Cambridge University Press, 1996.

    24 -
    -M. Snir, S. Otto, S. Huss-Lederman, D. Walker, J. Dongarra, -MPI: The Complete Reference. Volume 1 - The MPI Core, second edition, -MIT Press, 1998.

    25 -
    -K. Stüben, -An Introduction to Algebraic Multigrid, -in A. Schüller, U. Trottenberg, C. Oosterlee, Multigrid, -Academic Press, 2001.

    26 -
    -R. S. Tuminaro, C. Tong, -Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines, -in J. Donnelley, editor, Proceedings of SuperComputing 2000, Dallas, 2000.

    27 -
    -P. Vanek, J. Mandel and M. Brezina, -Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems, -Computing, 56, 1996, 179-196. +Copyright © 1993, 1994, 1995, 1996, +Nikos Drakos, +Computer Based Learning Unit, University of Leeds. +
    +Copyright © 1997, 1998, 1999, +Ross Moore, +Mathematics Department, Macquarie University, Sydney.

    -

    - +The command line arguments were:
    + latex2html -local_icons -noaddress -dir ../../html userhtml.tex

    +The translation was initiated by Salvatore Filippone on 2017-04-21

    diff --git a/docs/html/node3.html b/docs/html/node3.html index 8c3a270f..c71fd541 100644 --- a/docs/html/node3.html +++ b/docs/html/node3.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
    - Next: Next: Code Distribution - Up: Up: userhtml - Previous: Previous: Contents -   Contents

    @@ -58,12 +58,12 @@ General Overview

    The MULTI-LEVEL DOMAIN DECOMPOSITION PARALLEL PRECONDITIONERS PACKAGE BASED ON -PSBLAS (MLD2P4) provides parallel Algebraic MultiGrid (AMG) and domain decomposition -preconditioners, designed to provide scalable and easy-to-use preconditioners -multi-level Schwarz preconditioners [25,23], -to be used in the iterative solutions of sparse linear systems: +PSBLAS (MLD2P4) provides parallel Algebraic MultiGrid (AMG) and Domain +Decomposition preconditioners (see, e.g., [2,27,25]), +to be used in the iterative solution of linear systems,

    @@ -86,26 +86,37 @@ Ax=b, where $A$ is a square, real or complex, sparse matrix. Multi-level preconditioners can be obtained by combining several AMG cycles (V, W, K) with -different smoothers (Jacobi, hybrid forward/backward Gauss-Seidel, block-Jacobi, additive Schwarz methods). -An algebraic approach is used to -generate a hierarchy of coarse-level matrices and operators, without -explicitly using any information on the geometry of the original problem, e.g., -the discretization of a PDE. The smoothed aggregation technique is applied -as algebraic coarsening strategy [1,27]. -Either exact or approximate solvers are available to solve the coarsest-level system. Specifically, -different versions of sparse LU factorizations from external packages, and native incomplete -LU factorizations and iterative block-Jacobi solvers can be used. -All smoothers can be also exploited as one-level preconditioners. + ALT="$A$"> is a square, real or complex, sparse matrix. The name of the package comes from its original implementation, containing +multi-level additive and hybrid Schwarz preconditioners, as well as one-level additive +Schwarz preconditioners. The current version extends the original plan by including +multi-level cycles and smoothers widely used in multigrid methods. + +

    +The multi-level preconditioners implemented in MLD2P4 are obtained by combining +AMG cycles with smoothers and coarsest-level solvers. The V-, W-, and +K-cycles [2,23] are available, which allow to define +almost all the preconditioners in the package, including the multi-level hybrid +Schwarz ones; a specific cycle is implemented to obained multi-level additive +Schwarz preconditioners. The Jacobi, hybridforward/backward Gauss-Seidel, block-Jacobi, and additive Schwarz methods +are available as smoothers. An algebraic approach is used to generate a hierarchy of +coarse-level matrices and operators, without explicitly using any information on the +geometry of the original problem, e.g., the discretization of a PDE. To this end, +the smoothed aggregation technique [1,29] +is applied. Either exact or approximate solvers can be used on the coarsest-level +system. Specifically, different sparse LU factorizations from external +packages, and native incomplete LU factorizations and Jacobi, hybrid Gauss-Seidel, +and block-Jacobi solvers are available. All smoothers can be also exploited as one-level +preconditioners.

    MLD2P4 is written in Fortran 2003, following an object-oriented design through the exploitation of features -such as abstract data type creation, functional overloading, and -dynamic memory management. -The parallel implementation is based on a Single Program Multiple Data +such as abstract data type creation, type extension, functional overloading, and +dynamic memory management. The parallel implementation is based on a Single Program Multiple Data (SPMD) paradigm. Single and double precision implementations of MLD2P4 are available for both the real and the complex case, which can be used through a single @@ -113,84 +124,81 @@ interface.

    MLD2P4 has been designed to implement scalable and easy-to-use -multilevel preconditioners in the context of the PSBLAS -(Parallel Sparse BLAS) computational framework [18,17]. -PSBLAS provides basic linear algebra +multilevel preconditioners in the context of the PSBLAS (Parallel Sparse BLAS) +computational framework [19,18]. PSBLAS provides basic linear algebra operators and data management facilities for distributed sparse matrices, -as well as parallel Krylov solvers which can be coupled with the MLD2P4 preconditioners. +as well as parallel Krylov solvers which can be used with the MLD2P4 preconditioners. The choice of PSBLAS has been mainly motivated by the need of having a portable and efficient software infrastructure implementing ``de facto'' standard parallel sparse linear algebra kernels, to pursue goals such as performance, portability, modularity ed extensibility in the development of the preconditioner package. On the other hand, the implementation of MLD2P4 has led to some revisions and extentions of the original PSBLAS kernels. -The inter-process comunication required -by MLD2P4 is encapsulated into the PSBLAS routines, except few cases where -MPI [24] is explicitly called É ancora cosi???. Therefore, MLD2P4 can be run on any parallel -machine where PSBLAS and MPI implementations are available. +The inter-process comunication required by MLD2P4 is encapsulated +in the PSBLAS routines;therefore, MLD2P4 can be run on any parallel machine where PSBLAS +implementations are available.

    -MLD2P4 has a layered and modular software architecture where three main layers can be identified. -The lower layer consists of the PSBLAS kernels, the middle one implements +MLD2P4 has a layered and modular software architecture where three main layers can be +identified. The lower layer consists of the PSBLAS kernels, the middle one implements the construction and application phases of the preconditioners, and the upper one provides a uniform interface to all the preconditioners. This architecture allows for different levels of use of the package: -few black-box routines at the upper layer allow non-expert users to easily -build any preconditioner available in MLD2P4 and to apply it within a PSBLAS Krylov solver; -facilities are also available that allow more expert users to extend the set of smoothers -and solvers for building new versions of preconditioners. +few black-box routines at the upper layer allow all users to easily +build and apply any preconditioner available in MLD2P4; +facilities are also available allowing expert users to extend the set of smoothers +and solvers for building new versions of the preconditioners (see +Section 7).

    -We note that the user interface of MLD2P4 2.1 (Perche 2.1 e non 2.0???...Ricordarsi di cambiare il configure) -has been extended with respect to the previous versions -in order to separate the construction -of the multi-level hierarchy from the construction of the smoothers and solvers, and to allow for more flexibility -at each level. -The software architecture described in [8] has significantly evolved too, in order to fully exploit the -Fortran 2003 features implemented in PSBLAS 3. +We note that the user interface of MLD2P4 2.1 has been extended with respect to the +previous versions in order to separate the construction of the multi-level hierarchy from +the construction of the smoothers and solvers, and to allow for more flexibility +at each level. The software architecture described in [9] has significantly +evolved too, in order to fully exploit the Fortran 2003 features implemented in PSBLAS 3. However, compatibility with previous versions has been preserved.

    -This guide is organized as follows. General information on the distribution of the source code -is reported in Section 2, while details on the configuration -and installation of the package are given in Section 3. A short description of -the preconditioners implemented in MLD2P4 is provided -in Section 4, to help the users in choosing among them. -The basics for building and applying the preconditioners -with the Krylov solvers implemented in PSBLAS are reported in Section 5, where the -Fortran codes of a few sample programs are also shown. A reference guide for -the upper-layer routines of MLD2P4, that are the user interface, is provided -in Section 6. The error handling mechanism used by the package is briefly described -in Section 8. The copyright terms concerning the distribution and modification -of MLD2P4 are reported in Appendix A. +This guide is organized as follows. General information on the distribution of the source +code is reported in Section 2, while details on the configuration +and installation of the package are given in Section 3. A short description +of the preconditioners implemented in MLD2P4 is provided in Section 4, +to help the users in choosing among them. The basics for building and applying the +preconditioners with the Krylov solvers implemented in PSBLAS are reported +in Section 5, where the Fortran codes of a few sample programs +are also shown. A reference guide for the user interface routines is provided +in Section 6. Information on the extension of the package +through the addition of new smoothers and solvers is reported in Section 7. +The error handling mechanism used by the package +is briefly described in Section 8. The copyright terms concerning the +distribution and modification of MLD2P4 are reported in Appendix A.

    - next - up - previous - contents
    - Next: Next: Code Distribution - Up: Up: userhtml - Previous: Previous: Contents -   Contents diff --git a/docs/html/node4.html b/docs/html/node4.html index e26e8320..d4084a4c 100644 --- a/docs/html/node4.html +++ b/docs/html/node4.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
    - Next: Next: Configuring and Building MLD2P4 - Up: Up: userhtml - Previous: Previous: General Overview -   Contents

    @@ -67,7 +67,7 @@ where contact points for further information can be also found.

    The software is available under a modified BSD license, as specified -in Appendix A; please note that some of the optional +in Appendix A; please note that some of the optional third party libraries may be licensed under a different and more stringent license, most notably the GPL, and this should be taken into account when treating derived works. diff --git a/docs/html/node5.html b/docs/html/node5.html index 94bdefd2..41b1e2f3 100644 --- a/docs/html/node5.html +++ b/docs/html/node5.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
    - Next: Next: Prerequisites - Up: Up: userhtml - Previous: Previous: Code Distribution -   Contents

    @@ -57,7 +57,7 @@ Configuring and Building MLD2P4 In order to build MLD2P4 it is necessary to set up a Makefile with appropriate -values for your system; this is done by means of the configure +system-dependent variables; this is done by means of the configure script. The distribution also includes the autoconf and automake sources employed to generate the script, but usually this is not needed to build the software. @@ -79,15 +79,15 @@ real and complex data, in both single and double precision. Subsections

    diff --git a/docs/html/node6.html b/docs/html/node6.html index e2e63023..a4b459c5 100644 --- a/docs/html/node6.html +++ b/docs/html/node6.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
    - Next: Next: Optional third party libraries - Up: Up: Configuring and Building MLD2P4 - Previous: Previous: Configuring and Building MLD2P4 -   Contents

    @@ -60,13 +60,13 @@ The following base libraries are needed:
    BLAS
    [12,13,20] Many vendors provide optimized versions + HREF="node27.html#blas3">13,14,21] Many vendors provide optimized versions of BLAS; if no vendor version is available for a given platform, the ATLAS software (math-atlas.sourceforge.net/) + HREF="math-atlas.sourceforge.net">math-atlas.sourceforge.net) may be employed. The reference BLAS from Netlib (www.netlib.org/blas) are meant to define the standard @@ -79,24 +79,24 @@ The following base libraries are needed: experience is that configuring ATLAS for building full LAPACK does not work in the correct way. Our advice is first to download the LAPACK tarfile from www.netlib.org/lapac and install it independently of ATLAS. In this case, + HREF="www.netlib.org/lapack">www.netlib.org/lapack and install it independently of ATLAS. In this case, you need to modify the OPTS and NOOPT definitions for including -fPIC compilation option in the make.inc file of the LAPACK library.
    MPI
    [19,24] A version of MPI is available on most + HREF="node27.html#MPI2">20,26] A version of MPI is available on most high-performance computing systems.
    PSBLAS
    [16,18] Parallel Sparse BLAS (PSBLAS) is + HREF="node27.html#PSBLASGUIDE">17,19] Parallel Sparse BLAS (PSBLAS) is available from www.ce.uniroma2.it/psblas; version - 3.4.0 (or later) is required. Indeed, all the prerequisites + 3.5.0 (or later) is required. Indeed, all the prerequisites listed so far are also prerequisites of PSBLAS.
    @@ -108,26 +108,26 @@ compiler as MLD2P4.

    - next - up - previous - contents
    - Next: Next: Optional third party libraries - Up: Up: Configuring and Building MLD2P4 - Previous: Previous: Configuring and Building MLD2P4 -   Contents diff --git a/docs/html/node7.html b/docs/html/node7.html index 4316abfd..7c5a9b49 100644 --- a/docs/html/node7.html +++ b/docs/html/node7.html @@ -26,33 +26,34 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
    - Next: Next: Configuration options - Up: Up: Configuring and Building MLD2P4 - Previous: Previous: Prerequisites -   Contents

    -

    -Optional third party libraries +

    +
    +Optional third party libraries

    @@ -64,73 +65,74 @@ for multi-level preconditioners may change to reflect their presence.

    UMFPACK
    [9] + HREF="node27.html#UMFPACK">10] A sparse LU factorization package included in the SuiteSparse library, available from faculty.cse.tamu.edu/davis/suitesparse.html; it provides sequential factorization and triangular system solution for double - precision real and complex data. We tested - version 4.5.4. Note that for configuring SuiteSparse you should provide the right -path to the BLAS and LAPACK libraries in the SuiteSparse_config/SuiteSparse_config.mk file. + precision real and complex data. We tested version 4.5.4 of SuiteSparse. + Note that for configuring SuiteSparse you should provide the right path to the BLAS + and LAPACK libraries in the SuiteSparse_config/SuiteSparse_config.mk file.
    MUMPS
    [10] + HREF="node27.html#MUMPS">11] A sparse LU factorization package available from mumps.enseeiht.fr/; + HREF="mumps.enseeiht.fr">mumps.enseeiht.fr; it provides sequential and parallel factorizations and triangular system solution for single and double precision, real and complex data. We tested versions 4.10.0 and version 5.0.1.
    SuperLU
    [11] + HREF="node27.html#SUPERLU">12] A sparse LU factorization package available from crd.lbl.gov/~xiaoye/SuperLU/; it provides sequential factorization and triangular system solution for single and double precision, real and complex data. We tested version 4.3 and 5.0. If you installed BLAS from -ATLAS, remember to define the BLASLIB variable in the make.inc file. + ATLAS, remember to define the BLASLIB variable in the make.inc file.
    SuperLU_Dist
    [21] + HREF="node27.html#SUPERLUDIST">22] A sparse LU factorization package available from the same site as SuperLU; it provides parallel factorization and triangular system solution for double precision real and complex data. We tested version 3.3 and 4.2. If you installed BLAS from -ATLAS, remember to define the BLASLIB variable in the make.inc file and -to add the -std=c99 option to the C compiler options. -Note that this library requires the ParMETIS -library for parallel graph partitioning and fill-reducing matrix ordering available from --std=c99 option to the C compiler options. + Note that this library requires the ParMETIS + library for parallel graph partitioning and fill-reducing matrix ordering, available from + glaros.dtc.umn.edu/gkhome/metis/parmetis/overview. +
    +

    -

    -
    +
    - next - up - previous - contents
    - Next: Next: Configuration options - Up: Up: Configuring and Building MLD2P4 - Previous: Previous: Prerequisites -   Contents diff --git a/docs/html/node8.html b/docs/html/node8.html index a1ec2719..b88c47f0 100644 --- a/docs/html/node8.html +++ b/docs/html/node8.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
    - Next: Next: Bug reporting - Up: Up: Configuring and Building MLD2P4 - Previous: Previous: Optional third party libraries -   Contents

    @@ -56,11 +56,8 @@ Configuration options

    -CONTROLLARE HELP DEL CONFIGURE: Versione MLD2P4, Versione PSBLAS, Influential Environmental Variables??? - -

    -To build MLD2P4 the first step is to use the configure script -in the main directory to generate the necessary makefile(s). +In order to build MLD2P4, the first step is to use the configure script +in the main directory to generate the necessary makefile.

    As a minimal example consider the following: @@ -74,7 +71,7 @@ be specified with an absolute path). The full set of options may be looked at by issuing the command ./configure --help, which produces: 

    -`configure' configures MLD2P4 2.0 to adapt to many kinds of systems.
+`configure' configures MLD2P4 2.1 to adapt to many kinds of systems.
 
 Usage: ./configure [OPTION]... [VAR=VALUE]...
 
@@ -128,27 +125,55 @@ Fine tuning of the installation directories:
   --pdfdir=DIR            pdf documentation [DOCDIR]
   --psdir=DIR             ps documentation [DOCDIR]
 
+Program names:
+  --program-prefix=PREFIX            prepend PREFIX to installed program names
+  --program-suffix=SUFFIX            append SUFFIX to installed program names
+  --program-transform-name=PROGRAM   run sed PROGRAM on installed program names
+
 Optional Features:
   --disable-option-checking  ignore unrecognized --enable/--with options
   --disable-FEATURE       do not include FEATURE (same as --enable-FEATURE=no)
   --enable-FEATURE[=ARG]  include FEATURE [ARG=yes]
+  --disable-dependency-tracking  speeds up one-time build
+  --enable-dependency-tracking   do not reject slow dependency extractors
   --enable-serial         Specify whether to enable a fake mpi library to run
                           in serial mode.
+  --enable-long-integers  Specify usage of 64 bits integers.
 
 Optional Packages:
   --with-PACKAGE[=ARG]    use PACKAGE [ARG=yes]
   --without-PACKAGE       do not use PACKAGE (same as --with-PACKAGE=no)
   --with-psblas=DIR       The install directory for PSBLAS, for example,
-                          --with-psblas=/opt/packages/psblas-3.3
+                          --with-psblas=/opt/packages/psblas-3.5
   --with-psblas-incdir=DIR
                           Specify the directory for PSBLAS includes.
   --with-psblas-libdir=DIR
                           Specify the directory for PSBLAS library.
+  --with-ccopt            additional CCOPT flags to be added: will prepend
+                          to CCOPT
+  --with-fcopt            additional FCOPT flags to be added: will prepend
+                          to FCOPT
+  --with-libs             List additional link flags here. For example,
+                          --with-libs=-lspecial_system_lib or
+                          --with-libs=-L/path/to/libs
+  --with-clibs            additional CLIBS flags to be added: will prepend
+                          to CLIBS
+  --with-flibs            additional FLIBS flags to be added: will prepend
+                          to FLIBS
+  --with-library-path     additional LIBRARYPATH flags to be added: will
+                          prepend to LIBRARYPATH
+  --with-include-path     additional INCLUDEPATH flags to be added: will
+                          prepend to INCLUDEPATH
+  --with-module-path      additional MODULE_PATH flags to be added: will
+                          prepend to MODULE_PATH
   --with-extra-libs       List additional link flags here. For example,
                           --with-extra-libs=-lspecial_system_lib or
                           --with-extra-libs=-L/path/to/libs
-  --with-mumps=LIBNAME    Specify the libname for MUMPS. Default: "-lsmumps
-                          -ldmumps -lcmumps -lzmumps -lmumps_common -lpord"
+  --with-blas=<lib>       use BLAS library <lib>
+  --with-blasdir=<dir>    search for BLAS library in <dir>
+  --with-lapack=<lib>     use LAPACK library <lib>
+  --with-mumps=LIBNAME    Specify the libname for MUMPS. Default: autodetect
+                          with minimum "-lmumps_common -lpord"
   --with-mumpsdir=DIR     Specify the directory for MUMPS library and
                           includes. Note: you will need to add auxiliary
                           libraries with --extra-libs; this depends on how
@@ -194,24 +219,23 @@ Some influential environment variables:
   CFLAGS      C compiler flags
   CPPFLAGS    C/C++/Objective C preprocessor flags, e.g. -I<include dir> if
               you have headers in a nonstandard directory <include dir>
-  CPP         C preprocessor
   MPICC       MPI C compiler command
-  F77         Fortran 77 compiler command
-  FFLAGS      Fortran 77 compiler flags
-  MPIF77      MPI Fortran 77 compiler command
   MPIFC       MPI Fortran compiler command
+  CPP         C preprocessor
 
 Use these variables to override the choices made by `configure' or to help
 it to find libraries and programs with nonstandard names/locations.
 
 Report bugs to <bugreport@mld2p4.it>.
    -For instance, if a user has built and installed PSBLAS 3.4 under the + +

    +For instance, if a user has built and installed PSBLAS 3.5 under the /opt directory and is using the SuiteSparse package (which includes UMFPACK), then MLD2P4 might be configured with: 

    - ./configure --with-psblas=/opt/psblas-3.4/ \
+ ./configure --with-psblas=/opt/psblas-3.5/ \
  --with-umfpackincdir=/usr/include/suitesparse/
    Once the configure script has completed execution, it will have @@ -223,7 +247,9 @@ install directory under the name Make.inc.MLD2P4. To use the MUMPS solver package, the user has to add the appropriate options to the configure script; by default we are looking for the libraries --ldmumps -lsmumps -lzmumps -lzmumps -mumps_common -lpord. +-ldmumps -lsmumps -lzmumps -mumps_common -lpord. +Pasqua, c'era due volte lzmumps. L'ho eliminato, ma poi mi e' venuto +il dubbio che il secondo lzmumps dovesse essere modificato. MUMPS often uses additional packages such as ScaLAPACK, ParMETIS, SCOTCH, as well as enabling OpenMP; in such cases it is necessary to add linker options with the --with-extra-libs configure option. @@ -238,26 +264,26 @@ followed (optionally) by make install
    - next - up - previous - contents
    - Next: Next: Bug reporting - Up: Up: Configuring and Building MLD2P4 - Previous: Previous: Optional third party libraries -   Contents diff --git a/docs/html/node9.html b/docs/html/node9.html index 90036311..cd51404d 100644 --- a/docs/html/node9.html +++ b/docs/html/node9.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
    - Next: Next: Example and test programs - Up: Up: Configuring and Building MLD2P4 - Previous: Previous: Configuration options -   Contents

    @@ -54,7 +54,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds

    Bug reporting

    -If you find any bugs in our codes, please let us know at (DECIDERE A CHI FARE IL BUG REPORTING) +If you find any bugs in our codes, please let us know at bugreport@mld2p4.it @@ -62,7 +62,8 @@ If you find any bugs in our codes, please let us know at (DECIDERE A CHI FARE IL ; be aware that the amount of information needed to reproduce a problem in a parallel -program may vary quite a lot. +program may vary quite a lot. A chi va fatto il bug reporting? La +mail inviata a questo indirizzo non viene mai letta.
    diff --git a/docs/html/userhtml.html b/docs/html/userhtml.html index 3655cc3c..3dcd3433 100644 --- a/docs/html/userhtml.html +++ b/docs/html/userhtml.html @@ -23,18 +23,18 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next up previous - contents
    - Next: Next: Abstract -   Contents

    @@ -77,76 +77,74 @@ Feb. 28, 2017
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[(iterativ)28(e)-334(soluti)1(on)-334(of)-333(linear)-333(systems)-1(,)]TJ/F22 10.9091 Tf 186.98 -26.681 Td [(Ax)]TJ/F15 10.9091 Tf 17.447 0 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(b;)]TJ 0 g 0 G /F15 10.9091 Tf 182.57 0 Td [(\0501\051)]TJ 0 g 0 G - -398.512 -23.028 Td [(where)]TJ/F22 10.9091 Tf 31.223 0 Td [(A)]TJ/F15 10.9091 Tf 11.495 0 Td [(is)-304(a)-304(squar)1(e)-1(,)-309(real)-304(or)-304(complex,)-309(sparse)-304(matrix.)-435(Mul)1(ti-le)-1(v)28(el)-303(preconditioners)-304(can)-304(b)-28(e)]TJ -42.718 -13.549 Td [(obtained)-269(b)28(y)-269(com)27(b)1(ining)-269(se)-1(v)28(eral)-269(AMG)-269(cycles)-269(\050V,)-269(W,)-269(K\051)-269(with)-269(di\013eren)28(t)-269(smo)-28(others)-269(\050Jacobi,)]TJ 0 -13.549 Td [(h)28(ybrid)-369(forw)28(ard/bac)28(kw)27(ard)-369(Gauss-Seidel,)-378(blo)-28(c)28(k-Jacobi,)-378(additiv)28(e)-369(Sc)28(h)28(w)27(arz)-369(metho)-28(ds\051.)-551(An)]TJ 0 -13.55 Td 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[(from)-416(its)-416(original)-415(im)-1(p)1(lem)-1(en)28(tation,)-436(con)28(taining)-416(m)28(ulti-lev)27(el)-415(additiv)28(e)-416(and)-416(h)28(ybrid)-416(Sc)28(h)27(w)28(arz)]TJ 0 -13.55 Td [(preconditioners,)-514(as)-478(w)28(ell)-478(as)-478(one-lev)27(el)-478(additiv)28(e)-478(Sc)28(h)28(w)28(arz)-478(preconditioners.)-879(The)-478(curren)28(t)]TJ 0 -13.549 Td [(v)28(ersion)-351(extends)-350(the)-351(original)-350(plan)-350(b)27(y)-350(including)-350(m)27(u)1(lti-lev)27(el)-350(cycles)-351(and)-350(smo)-28(others)-350(w)-1(i)1(dely)]TJ 0 -13.549 Td [(used)-333(in)-334(m)28(ultigrid)-333(metho)-28(ds.)]TJ 16.937 -14.274 Td [(The)-248(m)28(ulti-lev)28(el)-247(preconditioners)-248(implemen)28(ted)-248(in)-247(MLD2P4)-248(are)-247(obtained)-247(b)27(y)-247(com)28(bining)]TJ -16.937 -13.549 Td [(AMG)-263(cycles)-264(with)-263(smo)-28(others)-263(and)-264(coarsest-lev)28(el)-263(solv)27(ers.)-421(The)-263(V-,)-278(W)1(-)-1(,)-277(and)-263(K-cycles)-264([)]TJ 1 0 0 rg 1 0 0 RG - [1(1)]TJ + [(2)]TJ 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-241(27)]TJ + [-263(23)]TJ 0 g 0 G - [(].)-414(Either)-240(exact)-241(or)-241(appro)28(ximate)-241(solv)28(ers)-241(are)-241(a)28(v)56(ailable)-241(to)-241(solv)28(e)-241(the)]TJ 0 -13.549 Td [(coarsest-lev)28(el)-403(system)-1(.)-653(Sp)-27(ec)-1(i\014)1(c)-1(al)1(ly)83(,)-420(di\013eren)27(t)-403(v)28(ersions)-403(of)-403(sparse)-403(LU)-403(f)1(ac)-1(t)1(orizations)-403(from)]TJ 0 -13.55 Td [(external)-368(p)1(ac)27(k)56(ages,)-376(and)-368(nativ)28(e)-367(incomplete)-368(LU)-368(f)1(ac)-1(tor)1(iz)-1(ati)1(ons)-368(and)-367(iterativ)27(e)-367(blo)-28(c)28(k-Jacobi)]TJ 0 -13.549 Td [(solv)28(ers)-334(can)-333(b)-28(e)-333(used.)-445(Al)1(l)-334(smo)-28(others)-333(can)-333(b)-28(e)-333(also)-334(exploited)-333(as)-333(one-lev)27(el)-333(preconditioners.)]TJ 16.937 -14.407 Td [(MLD2P4)-432(is)-431(written)-432(in)-432(F)84(ortran)-432(2003,)-456(follo)28(wing)-432(an)-432(ob)-55(ject-orien)28(ted)-432(design)-432(through)]TJ -16.937 -13.549 Td [(the)-305(exploitation)-304(of)-305(features)-305(suc)28(h)-305(as)-305(abstract)-304(data)-305(t)28(yp)-28(e)-305(creation,)-310(functional)-305(o)28(v)28(erloading,)]TJ 0 -13.549 Td [(and)-360(dynamic)-360(memory)-360(managemen)28(t.)-525(The)-360(parallel)-360(implemen)28(tation)-360(is)-360(based)-360(on)-360(a)-360(Single)]TJ 0 -13.55 Td [(Program)-388(Mu)1(ltiple)-388(Data)-388(\050SP)1(MD\051)-388(paradigm.)-607(Single)-388(and)-387(double)-388(precision)-387(implemen)28(ta-)]TJ 0 -13.549 Td [(tions)-390(of)-389(MLD2P4)-390(are)-389(a)28(v)55(ailable)-389(for)-390(b)-28(oth)-389(the)-390(real)-389(and)-390(the)-389(c)-1(omplex)-389(case,)-404(whic)28(h)-390(can)-389(b)-28(e)]TJ 0 -13.549 Td [(used)-333(through)-334(a)-333(single)-333(in)28(terface.)]TJ 16.937 -14.407 Td [(MLD2P4)-229(has)-230(b)-28(een)-229(designed)-229(to)-230(implemen)28(t)-230(scalable)-229(and)-229(easy-to-use)-230(m)28(ultilev)28(el)-230(precon-)]TJ -16.937 -13.55 Td [(ditioners)-349(in)-349(the)-350(con)28(text)-349(of)-349(the)-349(PSBLAS)-349(\050P)27(arallel)-349(Sparse)-349(BLAS\051)-349(computational)-349(frame-)]TJ 0 -13.549 Td [(w)28(ork)-360([)]TJ + [(])]TJ 0 -13.549 Td [(are)-392(a)28(v)56(ailable,)-407(whic)28(h)-392(allo)28(w)-392(to)-391(de\014ne)-392(almost)-392(all)-391(the)-392(preconditioners)-392(in)-392(th)1(e)-392(pac)28(k)55(age,)-406(in-)]TJ 0 -13.549 Td [(cluding)-317(the)-318(m)28(ulti-lev)28(el)-317(h)27(yb)1(rid)-318(Sc)28(h)28(w)28(arz)-318(ones;)-322(a)-318(sp)-28(eci\014c)-317(cycle)-317(is)-318(implemen)28(ted)-317(to)-318(obained)]TJ 0 -13.549 Td [(m)28(ulti-lev)28(el)-462(additiv)28(e)-462(Sc)28(h)28(w)28(arz)-462(preconditioners.)-829(The)-461(Jacobi,)-494(h)28(ybrid)-461(forw)28(ard/bac)27(kw)28(ard)]TJ 0 -13.55 Td [(Gauss-Seidel,)-366(blo)-27(c)27(k-Jacobi,)-365(and)-359(additiv)28(e)-360(Sc)28(h)28(w)28(arz)-359(m)-1(eth)1(o)-28(ds)-359(are)-360(a)28(v)56(ailable)-359(as)-360(smo)-27(others.)]TJ 0 -13.549 Td [(An)-279(algebraic)-279(appr)1(oac)27(h)-279(i)1(s)-279(used)-279(to)-279(generate)-279(a)-279(hierarc)28(h)28(y)-279(of)-279(coarse-lev)28(el)-279(matrices)-279(and)-278(op)-28(er-)]TJ 0 -13.549 Td [(ators,)-283(without)-270(explicitly)-270(using)-270(an)28(y)-271(inf)1(ormation)-271(on)-270(the)-270(geometry)-270(of)-270(the)-271(original)-270(problem,)]TJ 0 -13.549 Td [(e.g.,)-256(the)-237(discretization)-237(of)-237(a)-237(PDE.)-237(T)84(o)-237(this)-237(end,)-256(the)-237(smo)-28(othed)-237(aggregation)-237(tec)28(hnique)-237([)]TJ 1 0 0 rg 1 0 0 RG - [(18)]TJ + [(1)]TJ 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-360(17)]TJ + [-236(29)]TJ 0 g 0 G - [(].)-524(PSBLAS)-359(pro)28(vides)-360(basic)-360(linear)-360(algebra)-360(op)-27(erators)-360(and)-360(data)-360(managemen)28(t)]TJ 0 -13.549 Td [(facilities)-304(for)-304(distributed)-304(sparse)-305(matrices,)-310(as)-304(w)28(ell)-305(as)-304(parallel)-304(Krylo)28(v)-304(solv)27(ers)-304(whic)28(h)-304(can)-304(b)-28(e)]TJ 0 -13.549 Td [(coupled)-405(with)-405(the)-405(MLD2P)1(4)-405(preconditioners.)-659(The)-405(c)27(hoi)1(c)-1(e)-404(of)-405(PSBLAS)-405(has)-405(b)-28(een)-405(mainly)]TJ 0 -13.549 Td [(motiv)56(ated)-431(b)27(y)-431(the)-431(need)-431(of)-430(ha)27(ving)-431(a)-431(p)-27(ortable)-431(and)-431(e\016cien)28(t)-431(soft)27(w)28(are)-431(infrastructure)-431(im-)]TJ 0 -13.55 Td [(plemen)28(ting)-386(\134de)-385(facto")-386(standard)-386(parallel)-385(sparse)-386(linear)-385(algebra)-386(k)28(ernels,)-399(to)-386(pursue)-385(goals)]TJ 0 -13.549 Td [(suc)28(h)-315(as)-316(p)-27(e)-1(r)1(formance,)-319(p)-28(ortabilit)28(y)83(,)-319(mo)-27(dularit)28(y)-316(ed)-315(extensibilit)28(y)-315(in)-315(the)-316(dev)28(elopmen)28(t)-315(of)-315(the)]TJ 0 -13.549 Td [(preconditioner)-306(pac)28(k)55(age.)-435(On)-306(the)-306(other)-306(hand,)-312(the)-306(implemen)28(tation)-306(of)-306(MLD2P4)-306(has)-306(led)-306(to)]TJ 0 -13.549 Td [(some)-287(revisions)-287(and)-286(exte)-1(n)28(tions)-286(of)-287(the)-287(original)-287(PSBLAS)-286(k)27(ern)1(e)-1(ls.)-428(The)-287(in)28(ter-pro)-28(cess)-287(com)28(u-)]TJ 0 -13.549 Td [(nication)-395(required)-396(b)28(y)-396(MLD2P)1(4)-396(is)-396(encapsulated)-395(in)28(to)-396(the)-395(PSBLAS)-396(routin)1(e)-1(s,)-411(except)-395(few)]TJ 0 -13.55 Td [(cases)-359(where)-359(MPI)-359([)]TJ + [(])]TJ 0 -13.549 Td [(is)-377(applied.)-575(Either)-376(exact)-377(or)-377(appro)28(ximate)-377(solv)28(ers)-377(can)-377(b)-28(e)-377(used)-376(on)-377(the)-377(coarsest-lev)28(el)-377(sys-)]TJ 0 -13.55 Td [(tem.)-441(Sp)-27(eci\014cally)83(,)-324(di\013eren)28(t)-322(sparse)-322(LU)-322(factorizations)-321(from)-322(external)-322(pac)28(k)55(ages,)-324(and)-321(nativ)27(e)]TJ 0 -13.549 Td [(incomplete)-285(LU)-285(factorizations)-285(and)-285(Jacobi,)-295(h)28(ybrid)-285(Gauss-Seidel,)-294(and)-285(blo)-28(c)28(k-Jacobi)-285(solv)28(ers)]TJ 0 -13.549 Td [(are)-333(a)27(v)56(ailable.)-444(All)-334(smo)-28(oth)1(e)-1(r)1(s)-334(can)-333(b)-28(e)-333(also)-334(exploited)-333(as)-333(one-lev)27(el)-333(preconditioners.)]TJ 16.937 -14.274 Td [(MLD2P4)-267(is)-267(written)-268(in)-267(F)84(ortran)-267(2003,)-281(follo)28(wing)-267(an)-267(ob)-56(ject-orien)28(ted)-267(design)-268(th)1(rough)-268(the)]TJ -16.937 -13.549 Td [(exploitation)-338(of)-337(features)-338(suc)28(h)-338(as)-337(abstract)-338(data)-338(t)28(yp)-28(e)-337(creation,)-339(t)28(yp)-28(e)-338(extension,)-338(functional)]TJ 0 -13.549 Td [(o)28(v)28(erloading,)-326(and)-325(dynamic)-324(memory)-325(managemen)28(t.)-441(The)-325(parallel)-324(implemen)28(tation)-325(is)-324(based)]TJ 0 -13.549 Td [(on)-424(a)-424(Single)-424(Program)-424(Multiple)-424(Data)-424(\050SPMD\051)-424(paradigm.)-717(Single)-424(and)-424(double)-424(precision)]TJ 0 -13.549 Td [(implemen)28(tations)-486(of)-486(MLD2P4)-486(are)-486(a)28(v)56(ailable)-486(for)-486(b)-27(oth)-486(the)-486(real)-486(and)-486(the)-486(complex)-485(cas)-1(e,)]TJ 0 -13.549 Td [(whic)28(h)-334(can)-333(b)-28(e)-333(used)-333(through)-333(a)-334(single)-333(in)28(terface.)]TJ 16.937 -14.274 Td [(MLD2P4)-229(has)-230(b)-28(een)-229(designed)-229(to)-230(implemen)28(t)-230(scalable)-229(and)-229(easy-to-use)-230(m)28(ultilev)28(el)-230(precon-)]TJ -16.937 -13.549 Td [(ditioners)-349(in)-349(the)-350(con)28(text)-349(of)-349(the)-349(PSBLAS)-349(\050P)27(arallel)-349(Sparse)-349(BLAS\051)-349(computational)-349(frame-)]TJ 0 -13.549 Td [(w)28(ork)-360([)]TJ 1 0 0 rg 1 0 0 RG - [(24)]TJ + [(19)]TJ +0 g 0 G + [(,)]TJ +1 0 0 rg 1 0 0 RG + [-360(18)]TJ 0 g 0 G - [(])-359(is)-359(explicitly)-359(called)]TJ/F43 10.9091 Tf 196.637 2.637 Td [(\023)]TJ -0.984 -2.637 Td [(E)-413(ancora)-413(cosi???)]TJ/F15 10.9091 Tf 91.483 0 Td [(.)-521(Therefore,)-366(MLD2P4)-358(can)]TJ -287.136 -13.549 Td [(b)-28(e)-272(run)-272(on)-272(an)28(y)-272(parallel)-272(mac)28(hine)-272(where)-272(PSBLAS)-272(and)-271(MPI)-272(implemen)27(tation)1(s)-273(ar)1(e)-272(a)27(v)56(ailable.)]TJ 16.937 -14.407 Td [(MLD2P4)-342(has)-341(a)-342(la)28(y)28(ered)-342(and)-342(mo)-27(dular)-342(soft)28(w)28(are)-342(arc)28(hitecture)-342(where)-342(th)1(re)-1(e)-341(main)-342(la)28(y)28(ers)]TJ -16.937 -13.549 Td [(can)-458(b)-28(e)-458(iden)28(ti\014ed.)-819(The)-458(lo)28(w)28(er)-458(la)27(y)28(er)-458(consists)-458(of)-458(the)-458(PSBLAS)-458(k)28(ernels,)-490(the)-458(middle)-458(one)]TJ 0 -13.549 Td [(implemen)28(ts)-458(the)-457(construction)-457(and)-458(application)-457(phases)-458(of)-457(the)-457(preconditioners,)-489(and)-457(the)]TJ 0 -13.55 Td [(upp)-28(er)-433(one)-433(pro)28(vides)-433(a)-433(uniform)-433(in)28(terface)-433(to)-434(all)-433(the)-433(precondition)1(e)-1(rs.)-743(This)-433(arc)27(hitecture)]TJ 0 -13.549 Td [(allo)28(ws)-413(for)-413(di\013eren)28(t)-413(lev)28(els)-413(of)-412(use)-413(of)-413(the)-413(pac)28(k)56(age:)-604(few)-412(blac)27(k-b)-27(o)27(x)-412(routines)-413(at)-413(the)-412(upp)-28(er)]TJ 0 -13.549 Td [(la)28(y)28(er)-249(allo)28(w)-249(non-exp)-28(ert)-249(users)-249(to)-248(eas)-1(il)1(y)-249(build)-249(an)28(y)-249(preconditioner)-249(a)28(v)56(ailable)-249(in)-249(MLD2P4)-248(and)]TJ + [(].)-524(PSBLAS)-359(pro)28(vides)-360(basic)-360(linear)-360(algebra)-360(op)-27(erators)-360(and)-360(data)-360(managemen)28(t)]TJ 0 -13.55 Td [(facilities)-414(for)-414(distributed)-414(sparse)-414(matrices,)-434(as)-414(w)27(ell)-414(as)-414(parallel)-414(Krylo)28(v)-414(solv)28(ers)-414(whic)28(h)-414(can)]TJ 0 -13.549 Td [(b)-28(e)-401(used)-402(with)-401(the)-401(MLD2P4)-402(p)1(rec)-1(on)1(ditioners.)-649(The)-401(c)27(hoice)-401(of)-401(PSBLAS)-402(h)1(as)-402(b)-28(een)-401(mainly)]TJ 0 -13.549 Td [(motiv)56(ated)-431(b)27(y)-431(the)-431(need)-431(of)-430(ha)27(ving)-431(a)-431(p)-27(ortable)-431(and)-431(e\016cien)28(t)-431(soft)27(w)28(are)-431(infrastructure)-431(im-)]TJ 0 -13.549 Td [(plemen)28(ting)-386(\134de)-385(facto")-386(standard)-386(parallel)-385(sparse)-386(linear)-385(algebra)-386(k)28(ernels,)-399(to)-386(pursue)-385(goals)]TJ 0 -13.549 Td [(suc)28(h)-315(as)-316(p)-27(e)-1(r)1(formance,)-319(p)-28(ortabilit)28(y)83(,)-319(mo)-27(dularit)28(y)-316(ed)-315(extensibilit)28(y)-315(in)-315(the)-316(dev)28(elopmen)28(t)-315(of)-315(the)]TJ 0 -13.55 Td [(preconditioner)-403(pac)28(k)56(age.)-653(On)-402(the)-403(other)-403(hand,)-420(the)-402(implemen)27(tation)-402(of)-403(MLD2P4)-402(has)-403(led)]TJ 0 -13.549 Td [(to)-431(some)-431(revisions)-431(and)-431(exten)28(tions)-431(of)-431(the)-431(original)-431(PSBLAS)-431(k)28(ernels.)-737(The)-431(in)28(ter-pro)-28(cess)]TJ 0 -13.549 Td [(com)28(unication)-340(required)-339(b)28(y)-340(MLD2P4)-339(is)-340(encapsul)1(ate)-1(d)-339(in)-339(the)-340(PSBLAS)-339(routines;therefore,)]TJ 0 g 0 G 0 g 0 G ET endstream endobj -207 0 obj +217 0 obj << -/Length 4243 +/Length 5495 >> stream 0 g 0 G BT /F15 10.9091 Tf 86.4 740.002 Td [(2)]TJ/F41 10.9091 Tf 203.265 0 Td [(MLD2P4)-378(User)67('s)-378(and)-378(Ref)1(erence)-378(Guide)]TJ 0 g 0 G -/F15 10.9091 Tf -203.265 -35.866 Td [(to)-316(apply)-316(it)-316(within)-316(a)-316(PSBLAS)-316(Krylo)28(v)-316(solv)28(er;)]TJ/F43 10.9091 Tf 211.858 0 Td [(facilities)-364(are)-363(also)-364(a)32(v)64(ailable)-364(that)-363(allo)32(w)]TJ -211.858 -13.549 Td [(more)-438(exp)-32(ert)-439(users)-438(to)-439(extend)-438(the)-439(set)-438(of)-439(smo)-32(others)-438(and)-438(solv)31(ers)-438(for)-438(buil)-1(ding)]TJ 0 -13.549 Td [(new)-383(v)32(ersions)-384(of)-383(preconditioners.)]TJ/F15 10.9091 Tf 16.936 -13.549 Td [(W)83(e)-361(note)-361(that)-362(the)-361(user)-361(in)27(terface)-361(of)-361(MLD2P4)-362(2.1)-361(\050)]TJ/F43 10.9091 Tf 236.965 0 Td [(P)32(erc)32(he)-416(2.1)-415(e)-416(non)-415(2.0???...Ri-)]TJ -253.901 -13.549 Td [(cordarsi)-399(di)-399(cam)32(biare)-399(il)-398(con\014gure)]TJ/F15 10.9091 Tf 177.704 0 Td [(\051)-347(has)-346(b)-28(een)-347(extended)-347(with)-346(resp)-28(ect)-347(to)-347(the)-347(p)1(re)-1(v)1(ious)]TJ -177.704 -13.55 Td [(v)28(ersions)-298(in)-298(order)-298(to)-298(separate)-298(the)-298(construction)-298(of)-298(the)-298(m)27(ulti-lev)28(el)-298(hierarc)28(h)28(y)-298(from)-298(the)-298(con-)]TJ 0 -13.549 Td [(struction)-375(of)-376(the)-375(smo)-28(others)-375(and)-376(solv)28(ers,)-386(and)-375(to)-376(allo)28(w)-375(for)-376(more)-375(\015exibilit)28(y)-376(at)-375(eac)28(h)-376(lev)28(el.)]TJ 0 -13.549 Td [(The)-289(soft)28(w)27(are)-289(arc)28(hitecture)-289(desc)-1(r)1(ib)-28(ed)-289(in)-290([)]TJ +/F15 10.9091 Tf -203.265 -35.866 Td [(MLD2P4)-491(can)-491(b)-28(e)-491(run)-491(on)-491(an)28(y)-492(p)1(arallel)-492(mac)28(hine)-491(where)-491(PSBLAS)-491(implemen)28(tations)-491(are)]TJ 0 -13.549 Td [(a)28(v)55(ailable.)]TJ 16.936 -13.549 Td [(MLD2P4)-342(has)-341(a)-342(la)28(y)28(ered)-342(and)-342(mo)-27(dular)-342(soft)28(w)28(are)-342(arc)28(hitecture)-342(where)-342(three)-341(main)-342(la)28(y)28(ers)]TJ -16.936 -13.549 Td [(can)-458(b)-28(e)-458(iden)28(ti\014ed.)-819(The)-458(lo)28(w)28(er)-458(la)27(y)28(er)-458(consists)-458(of)-458(the)-458(PSBLAS)-458(k)28(ernels,)-490(the)-458(middle)-458(one)]TJ 0 -13.549 Td [(implemen)28(ts)-458(the)-457(construction)-457(and)-458(application)-457(phases)-457(of)-458(the)-457(preconditioners,)-489(and)-457(the)]TJ 0 -13.55 Td [(upp)-28(er)-433(one)-433(pro)28(vides)-433(a)-433(uniform)-433(in)28(terface)-433(to)-433(all)-434(th)1(e)-434(preconditi)1(oners)-1(.)-743(This)-433(arc)27(h)1(ite)-1(ctu)1(re)]TJ 0 -13.549 Td 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[(].)-566(The)-373(same)-374(reasoning)-374(can)-373(b)-28(e)-374(applied)-373(s)-1(t)1(arting)-374(from)-374(the)-373(coarse-)]TJ 0 -13.55 Td [(lev)28(el)-281(system,)-291(i.e.)-281(a)-280(coarse-s)-1(p)1(ac)-1(e)-280(correction)-281(can)-280(b)-28(e)-281(built)-280(from)-281(this)-280(sys)-1(t)1(e)-1(m,)-291(th)28(us)-280(obtaining)]TJ/F18 10.9091 Tf 0 -13.549 Td [(multi-level)]TJ/F15 10.9091 Tf 53.83 0 Td [(preconditioners.)]TJ -36.894 -14.011 Td [(It)-443(is)-444(w)28(orth)-443(noting)-444(that)-443(optimal)-444(p)1(rec)-1(on)1(ditioners)-444(do)-443(not)-444(necessarily)-443(corresp)-28(ond)-443(to)]TJ -16.936 -13.549 Td [(minim)28(um)-480(exec)-1(u)1(tion)-481(times.)-885(Indeed,)-517(to)-480(obtain)-480(e\013ectiv)28(e)-481(m)28(ulti-lev)28(el)-480(preconditioners)-480(a)]TJ 0 -13.549 Td [(tradeo\013)-381(b)-28(et)28(w)28(ee)-1(n)-381(optimalit)28(y)-381(of)-381(con)28(v)27(ergence)-381(and)-381(the)-381(cost)-382(of)-381(building)-381(and)-381(applying)-381(the)]TJ 0 -13.55 Td [(coarse-space)-384(corrections)-384(m)28(ust)-384(b)-28(e)-383(ac)27(hiev)28(ed.)-596(Th)1(e)-384(c)27(hoice)-383(of)-384(the)-384(n)28(um)28(b)-28(er)-384(of)-383(lev)27(els,)-396(i.e.)-384(of)]TJ 0 -13.549 Td [(the)-379(coarse-space)-379(corr)1(e)-1(ctions,)-390(al)1(s)-1(o)-378(a\013ects)-379(the)-379(e\013ectiv)28(eness)-379(of)-378(the)-379(preconditioners.)-580(One)]TJ 0 -13.549 Td [(more)-392(goal)-391(is)-392(to)-392(get)-391(con)28(v)27(ergence)-391(rates)-392(as)-392(less)-392(sensitiv)28(e)-392(as)-391(p)-28(ossible)-392(to)-391(v)55(ariations)-391(in)-392(the)]TJ 0 -13.549 Td [(matrix)-333(co)-28(e\016cien)28(ts.)]TJ 16.936 -14.012 Td [(Tw)28(o)-303(main)-303(approac)28(hes)-303(can)-303(b)-28(e)-303(used)-303(to)-303(build)-303(coarse-space)-303(corrections.)-434(The)-303(geometric)]TJ -16.936 -13.549 Td [(approac)28(h)-412(applies)-413(coarsening)-412(strategies)-413(b)1(as)-1(ed)-412(on)-412(the)-412(kno)27(wledge)-412(of)-412(some)-413(ph)28(ysical)-412(grid)]TJ 0 -13.549 Td [(asso)-28(ciated)-279(to)-280(the)-279(matrix)-280(and)-279(requires)-280(the)-279(user)-280(to)-280(d)1(e)-1(\014)1(ne)-280(grid)-279(transfer)-280(op)-28(erators)-279(from)-280(the)]TJ 0 -13.549 Td [(\014ne)-393(to)-394(the)-394(coarse)-393(lev)28(e)-1(l)1(s)-394(and)-394(vice)-393(v)28(ersa.)-626(Th)1(is)-394(ma)28(y)-394(result)-393(di\016cult)-394(for)-393(complex)-394(geome-)]TJ 0 -13.549 Td [(tries;)-464(furthermore,)-442(suitable)-420(one-lev)28(e)-1(l)-420(preconditioners)-420(ma)28(y)-421(b)-28(e)-420(required)-420(to)-421(get)-420(e\016cien)28(t)]TJ 0 -13.549 Td [(in)28(terpla)28(y)-381(b)-28(et)28(w)27(een)-381(\014ne)-381(and)-381(coarse)-381(lev)27(els,)-393(e.g.)-381(when)-381(matrices)-381(with)-381(highly)-381(v)55(arying)-381(co)-28(ef-)]TJ 0 -13.55 Td [(\014cien)28(ts)-417(are)-417(consid)1(e)-1(r)1(e)-1(d)1(.)-695(The)-416(alge)-1(b)1(raic)-417(approac)28(h)-417(builds)-416(coarse-s)-1(p)1(ac)-1(e)-416(corrections)-417(using)]TJ 0 -13.549 Td [(only)-442(matrix)-442(information.)-770(It)-442(p)-28(erforms)-442(a)-442(fully)-442(automatic)-442(coarsening)-442(and)-442(enforces)-443(th)1(e)]TJ +/F15 10.9091 Tf 34.09 0 Td [(con)28(tains)-380(a)-380(set)-380(of)-380(more)-380(sophi)1(s)-1(ticated)-379(examples)-380(that)-380(will)-380(allo)28(w)-380(the)-380(user,)-391(via)-380(the)]TJ -6.817 -13.549 Td [(input)-286(\014les)-287(in)-286(the)]TJ/F44 10.9091 Tf 80.438 0 Td [(runs)]TJ/F15 10.9091 Tf 26.033 0 Td [(sub)-28(directories,)-296(to)-286(exp)-28(erimen)28(t)-286(with)-287(the)-286(full)-287(ran)1(ge)-287(of)-286(precon-)]TJ -106.471 -13.549 Td [(ditioners)-333(implemen)28(ted)-334(in)-333(the)-333(pac)27(k)56(age.)]TJ -27.273 -22.516 Td [(The)]TJ/F44 10.9091 Tf 24.239 0 Td [(fileread)]TJ/F15 10.9091 Tf 51.269 0 Td [(directories)-500(con)28(tain)-500(sample)-499(programs)-500(that)-500(read)-499(sparse)-500(matrices)-500(from)]TJ -75.508 -13.549 Td [(\014les,)-295(according)-285(to)-285(the)-285(Matrix)-285(Mark)28(et)-285(or)-285(the)-285(Harw)27(ell-Bo)-27(eing)-286(storage)-285(format;)-301(the)]TJ/F44 10.9091 Tf 378.088 0 Td [(pdegen)]TJ/F15 10.9091 Tf -378.088 -13.549 Td [(programs)-416(generate)-415(matrices)-416(in)-416(full)-415(parallel)-416(mo)-28(de)-416(from)-415(the)-416(discretization)-416(of)-415(a)-416(sample)]TJ 0 -13.549 Td [(partial)-333(di\013eren)28(tial)-334(equation.)]TJ 0 g 0 G 0 g 0 G ET endstream endobj -305 0 obj +308 0 obj << -/Length 14867 +/Length 8161 >> stream 0 g 0 G @@ -1131,129 +1122,145 @@ stream BT /F41 10.9091 Tf 93.6 740.002 Td [(4)]TJ 0 g 0 G - [-378(Mul)67(ti-level)-378(Domain)-378(Decompo)1(siti)-1(o)1(n)-378(Ba)22(ck)22(gr)22(ound)]TJ/F15 10.9091 Tf 401.542 0 Td [(11)]TJ -0 g 0 G - -401.542 -35.866 Td [(in)28(terpla)28(y)-352(b)-28(et)28(w)28(een)-352(the)-352(\014ne)-351(and)-352(coarse)-352(lev)28(els)-352(b)28(y)-351(s)-1(u)1(itably)-352(c)28(ho)-28(osing)-352(the)-351(coarse)-352(space)-352(and)]TJ 0 -13.549 Td [(the)-333(coarse-to-\014ne)-334(in)28(terp)-28(olation)-333([)]TJ -1 0 0 rg 1 0 0 RG - [(25)]TJ + [-378(Mul)67(tigrid)-378(Ba)22(ck)22(gr)23(ound)]TJ/F15 10.9091 Tf 401.542 0 Td [(11)]TJ 0 g 0 G - [(].)]TJ 16.937 -13.643 Td [(MLD2P4)-255(uses)-255(a)-255(pu)1(re)-255(algebraic)-255(approac)28(h)-255(for)-255(building)-255(t)1(he)-255(sequence)-255(of)-255(coarse)-255(matrices)]TJ -16.937 -13.549 Td [(starting)-425(from)-425(the)-425(original)-425(matrix.)-720(The)-426(algebrai)1(c)-426(approac)28(h)-425(is)-425(based)-425(on)-425(the)]TJ/F18 10.9091 Tf 369.239 0 Td [(smo)51(othe)51(d)]TJ -369.239 -13.549 Td [(aggr)51(e)51(gation)]TJ/F15 10.9091 Tf 58.613 0 Td [(algorithm)-402([)]TJ +/F17 14.3462 Tf -401.542 -35.866 Td [(4)-1125(Multigrid)-375(Bac)31(kground)]TJ/F15 10.9091 Tf 0 -25.431 Td [(Multigrid)-467(preconditioners,)-500(coupled)-468(with)-467(Krylo)28(v)-467(iterativ)28(e)-467(solv)27(ers,)-500(are)-467(widely)-467(use)-1(d)-467(in)]TJ 0 -13.549 Td [(the)-374(par)1(allel)-374(solution)-373(of)-374(large)-374(and)-373(sparse)-374(linear)-373(systems,)-384(b)-28(ecause)-373(of)-374(their)-373(optimalit)28(y)-374(in)]TJ 0 -13.55 Td [(the)-411(solution)-411(of)-411(linear)-411(systems)-411(arising)-411(from)-411(the)-412(d)1(isc)-1(r)1(e)-1(tization)-411(of)-411(scalar)-411(elliptic)-411(P)28(artial)]TJ 0 -13.549 Td [(Di\013eren)28(tial)-331(Equations)-330(\050PDEs\051)-331(on)-330(regular)-331(grids.)-443(Optimalit)28(y)83(,)-331(also)-331(kn)1(o)27(wn)-330(as)-331(algorithmic)]TJ 0 -13.549 Td [(scalabilit)28(y)83(,)-404(is)-389(the)-390(prop)-28(ert)28(y)-390(of)-390(ha)28(ving)-389(a)-390(computational)-390(cost)-390(p)-27(er)-390(iteration)-390(that)-390(dep)-27(ends)]TJ 0 -13.549 Td [(linearly)-291(on)-292(the)-291(problem)-292(size,)-299(and)-292(a)-291(con)28(v)27(ergence)-291(rate)-292(that)-291(is)-291(indep)-28(enden)28(t)-292(of)-291(the)-292(prob)1(le)-1(m)]TJ 0 -13.549 Td [(size.)]TJ 16.937 -14.105 Td [(Multigrid)-304(preconditioners)-304(are)-304(based)-304(on)-304(a)-304(recursiv)28(e)-304(application)-304(of)-304(a)-304(t)28(w)28(o-grid)-304(pro)-28(cess)]TJ -16.937 -13.549 Td [(consisting)-391(of)-391(smo)-28(other)-391(iterations)-391(and)-392(a)-391(coarse-space)-391(\050or)-391(coarse-lev)27(el\051)-391(correction.)-618(The)]TJ 0 -13.55 Td [(smo)-28(others)-343(ma)27(y)-343(b)-28(e)-343(either)-343(basic)-344(iterativ)28(e)-343(m)-1(eth)1(o)-28(ds,)-346(suc)28(h)-344(as)-343(the)-344(Jacobi)-343(and)-343(Gauss-Seidel)]TJ 0 -13.549 Td [(ones,)-434(or)-414(more)-414(complex)-414(subspace-correction)-414(metho)-28(ds,)-434(suc)28(h)-414(as)-414(the)-414(Sc)28(h)27(w)28(arz)-414(ones.)-686(The)]TJ 0 -13.549 Td [(coarse-space)-304(correction)-304(consists)-303(of)-304(solving,)-310(in)-303(an)-304(appropriately)-303(c)27(hosen)-303(coarse)-304(space,)-310(the)]TJ 0 -13.549 Td [(residual)-297(equation)-296(asso)-28(ciated)-297(with)-296(the)-297(appro)28(ximate)-297(solution)-296(computed)-297(b)28(y)-297(th)1(e)-297(smo)-28(other,)]TJ 0 -13.549 Td [(and)-359(of)-359(using)-359(the)-359(solution)-359(of)-359(this)-359(equation)-359(to)-359(correct)-359(the)-359(previous)-359(appro)28(ximation.)-522(The)]TJ 0 -13.55 Td [(transfer)-467(of)-467(information)-468(b)-27(et)27(w)28(een)-467(the)-467(original)-468(\050\014)1(ne\051)-468(space)-467(and)-467(the)-468(coarse)-467(one)-467(is)-468(p)-27(er-)]TJ 0 -13.549 Td [(formed)-394(b)27(y)-394(using)-394(suitable)-395(restriction)-394(and)-394(prolongation)-395(op)-27(erators.)-628(The)-394(construction)-395(of)]TJ 0 -13.549 Td [(the)-410(coarse)-409(s)-1(p)1(ac)-1(e)-409(and)-410(the)-410(corresp)-27(onding)-410(transfer)-410(op)-27(erators)-410(is)-410(carried)-409(out)-410(b)28(y)-410(applying)]TJ 0 -13.549 Td [(a)-390(so-called)-391(coarsening)-390(algorithm)-390(to)-390(the)-390(s)-1(ystem)-390(matrix.)-615(Tw)28(o)-391(main)-390(approac)28(hes)-390(can)-390(b)-28(e)]TJ 0 -13.549 Td [(used)-370(to)-369(p)-28(erform)-369(c)-1(oar)1(s)-1(enin)1(g:)-517(the)-370(geometric)-370(approac)28(h,)-378(whic)27(h)-369(exploits)-370(the)-369(kno)28(wledge)-370(of)]TJ 0 -13.55 Td [(some)-361(ph)28(ysical)-361(grid)-361(ass)-1(o)-27(ciated)-361(with)-361(the)-362(matri)1(x)-362(and)-361(requi)1(res)-362(the)-361(user)-361(to)-361(de\014ne)-361(transfer)]TJ 0 -13.549 Td [(op)-28(erators)-348(from)-349(the)-348(\014ne)-349(to)-348(the)-349(coarse)-348(lev)27(el)-348(and)-349(vi)1(c)-1(e)-348(v)28(ersa,)-353(and)-348(the)-349(algebrai)1(c)-349(approac)28(h,)]TJ 0 -13.549 Td [(whic)28(h)-282(builds)-283(the)-282(coarse-space)-282(correc)-1(ti)1(on)-283(and)-282(the)-282(asso)-28(ciate)-282(transfer)-283(op)-27(erators)-283(using)-282(only)]TJ 0 -13.549 Td [(matrix)-369(inf)1(ormation.)-551(The)-368(\014rst)-369(approac)28(h)-369(ma)28(y)-369(b)-27(e)-369(di\016cult)-369(wh)1(e)-1(n)-368(the)-369(system)-368(c)-1(omes)-368(from)]TJ 0 -13.549 Td [(discretizations)-288(on)-288(complex)-287(ge)-1(ometries;)-303(fur)1(thermore,)-297(ad)-288(ho)-28(c)-288(one-lev)28(el)-288(smo)-28(others)-288(ma)28(y)-288(b)-27(e)]TJ 0 -13.549 Td [(required)-307(to)-307(get)-306(an)-307(e\016cien)27(t)-306(in)27(terpla)28(y)-307(b)-27(et)27(w)28(een)-307(\014ne)-307(and)-306(coarse)-307(lev)28(e)-1(ls,)-312(e.g.,)-312(when)-307(matrices)]TJ 0 -13.55 Td [(with)-363(hi)1(ghly)-363(v)56(arying)-363(co)-28(e\016cien)28(ts)-363(are)-362(considered.)-532(The)-363(second)-363(appr)1(oac)27(h)-362(p)-28(erforms)-363(a)-362(fully)]TJ 0 -13.549 Td [(automatic)-349(coarsening)-350(and)-349(enforces)-349(the)-349(in)27(t)1(e)-1(rp)1(la)27(y)-349(b)-28(et)28(w)28(een)-349(\014ne)-350(and)-349(coarse)-349(lev)27(el)-349(b)28(y)-349(suit-)]TJ 0 -13.549 Td [(ably)-313(c)28(ho)-28(osing)-313(the)-313(coarse)-313(space)-313(and)-313(the)-313(coarse-to-\014ne)-313(in)28(terp)-28(olation)-313(\050see,)-317(e.g.,)-317([)]TJ 1 0 0 rg 1 0 0 RG - [(1)]TJ + [(2)]TJ 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-402(27)]TJ -0 g 0 G - [(].)-651(A)-403(decoupled)-402(v)28(ersion)-402(of)-402(this)-403(algorithm)-402(is)-402(implemen)28(ted,)]TJ -58.613 -13.549 Td [(where)-347(the)-346(smo)-28(othed)-347(aggregation)-346(is)-347(applied)-347(lo)-27(cally)-347(to)-347(eac)28(h)-346(s)-1(u)1(bmatrix)-347([)]TJ -1 0 0 rg 1 0 0 RG - [(26)]TJ -0 g 0 G - [(].)-484(In)-347(the)-347(next)]TJ 0 -13.549 Td [(t)28(w)28(o)-249(subsections)-249(w)28(e)-249(pro)28(vide)-249(a)-248(brief)-249(description)-249(of)-248(the)-249(m)28(ulti-lev)28(e)-1(l)-248(Sc)28(h)27(w)28(arz)-249(pr)1(e)-1(cond)1(itioners)]TJ 0 -13.55 Td [(and)-389(of)-390(the)-389(smo)-28(othed)-390(aggregation)-389(tec)28(hnique)-390(as)-389(implemen)27(ted)-389(in)-390(MLD)1(2P4.)-613(F)83(or)-390(f)1(urther)]TJ 0 -13.549 Td [(details)-333(the)-334(reader)-333(is)-333(referred)-334(to)-333([)]TJ -1 0 0 rg 1 0 0 RG - [(2)]TJ + [-313(27)]TJ 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-333(3)]TJ + [-313(25)]TJ 0 g 0 G - [(,)]TJ + [(])]TJ 0 -13.549 Td [(for)-333(details.\051)]TJ 16.937 -14.105 Td [(MLD2P4)-329(uses)-330(a)-329(pure)-330(algebraic)-329(approac)28(h,)-331(b)1(as)-1(ed)-329(on)-329(the)-330(smo)-28(othed)-329(aggregation)-330(algo-)]TJ -16.937 -13.549 Td [(rithm)-298([)]TJ 1 0 0 rg 1 0 0 RG - [-334(4)]TJ + [(1)]TJ 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-333(8)]TJ + [-298(29)]TJ 0 g 0 G - [(,)]TJ + [(],)-305(for)-298(building)-298(the)-298(sequence)-298(of)-298(coarse)-298(matrices)-298(and)-298(transfer)-298(op)-28(erators,)-305(start-)]TJ 0 -13.549 Td [(ing)-306(from)-306(the)-305(original)-306(one.)-435(A)-306(decoupled)-306(v)28(ersion)-306(of)-306(this)-306(algori)1(thm)-306(is)-306(implemen)28(ted,)-312(where)]TJ 0 -13.55 Td [(the)-316(smo)-28(othed)-316(aggregation)-315(is)-316(applied)-316(lo)-28(cally)-316(to)-316(eac)28(h)-316(submatrix)-316([)]TJ 1 0 0 rg 1 0 0 RG - [-333(23)]TJ + [(28)]TJ 0 g 0 G - [(].)]TJ/F17 11.9552 Tf 0 -29.748 Td [(4.1)-1125(Multi-lev)31(el)-375(Sc)31(h)32(w)31(arz)-375(Preconditioners)]TJ/F15 10.9091 Tf 0 -20.777 Td [(The)-270(Multilev)28(el)-270(p)1(re)-1(cond)1(itioners)-270(implemen)28(ted)-270(in)-270(MLD2P4)-269(are)-270(obtained)-270(b)28(y)-269(com)27(bining)-269(AS)]TJ 0 -13.549 Td [(preconditioners)-315(with)-315(coarse-space)-315(corrections;)-321(therefore)-316(w)28(e)-315(\014rst)-315(pro)28(vide)-315(a)-315(sk)28(etc)27(h)-315(of)-315(the)]TJ 0 -13.549 Td [(AS)-333(preconditioners.)]TJ 16.937 -13.643 Td [(Giv)28(en)-303(th)1(e)-303(linear)-302(system)-303(\050)]TJ + [(].)-438(A)-316(brief)-316(description)]TJ 0 -13.549 Td [(of)-333(the)-333(AMG)-333(prec)-1(on)1(ditioners)-334(implemen)28(ted)-333(in)-333(MLD2P4)-333(is)-333(giv)27(en)-333(in)-333(Sections)]TJ 0 0 1 rg 0 0 1 RG - [(1)]TJ + [-333(4.1)]TJ 0 g 0 G - [(\051,)-308(where)]TJ/F22 10.9091 Tf 167.085 0 Td [(A)]TJ/F15 10.9091 Tf 11.212 0 Td [(=)-278(\050)]TJ/F22 10.9091 Tf 15.757 0 Td [(a)]TJ/F23 7.9701 Tf 5.767 -1.636 Td [(ij)]TJ/F15 10.9091 Tf 7.265 1.636 Td [(\051)]TJ/F25 10.9091 Tf 7.273 0 Td [(2)-278(<)]TJ/F23 7.9701 Tf 18.181 3.959 Td [(n)]TJ/F26 7.9701 Tf 5.139 0 Td [(\002)]TJ/F23 7.9701 Tf 6.586 0 Td [(n)]TJ/F15 10.9091 Tf 8.935 -3.959 Td [(is)-302(a)-303(nonsingular)-302(sparse)-302(matrix)]TJ -270.137 -13.549 Td [(with)-321(a)-321(symmetric)-321(nonzero)-321(pattern,)-324(let)]TJ/F22 10.9091 Tf 184.527 0 Td [(G)]TJ/F15 10.9091 Tf 11.607 0 Td [(=)-278(\050)]TJ/F22 10.9091 Tf 15.758 0 Td [(W)28(;)-167(E)]TJ/F15 10.9091 Tf 23.53 0 Td [(\051)-321(b)-28(e)-321(the)-321(adjacency)-321(graph)-321(of)]TJ/F22 10.9091 Tf 134.379 0 Td [(A)]TJ/F15 10.9091 Tf 8.182 0 Td [(,)-323(where)]TJ/F22 10.9091 Tf -377.983 -13.549 Td [(W)]TJ/F15 10.9091 Tf 15.419 0 Td [(=)]TJ/F25 10.9091 Tf 12.086 0 Td [(f)]TJ/F15 10.9091 Tf 5.455 0 Td [(1)]TJ/F22 10.9091 Tf 5.455 0 Td [(;)]TJ/F15 10.9091 Tf 4.848 0 Td [(2)]TJ/F22 10.9091 Tf 5.455 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(n)]TJ/F25 10.9091 Tf 30.79 0 Td [(g)]TJ/F15 10.9091 Tf 9.433 0 Td [(and)]TJ/F22 10.9091 Tf 21.555 0 Td [(E)]TJ/F15 10.9091 Tf 12.283 0 Td [(=)]TJ/F25 10.9091 Tf 12.086 0 Td [(f)]TJ/F15 10.9091 Tf 5.454 0 Td [(\050)]TJ/F22 10.9091 Tf 4.243 0 Td [(i;)-167(j)]TJ/F15 10.9091 Tf 13.723 0 Td [(\051)-330(:)]TJ/F22 10.9091 Tf 14.475 0 Td [(a)]TJ/F23 7.9701 Tf 5.767 -1.637 Td [(ij)]TJ/F25 10.9091 Tf 10.866 1.637 Td [(6)]TJ/F15 10.9091 Tf 0 0 Td [(=)-330(0)]TJ/F25 10.9091 Tf 17.541 0 Td [(g)]TJ/F15 10.9091 Tf 9.433 0 Td [(are)-365(the)-364(v)27(ertex)-364(se)-1(t)-364(and)-365(the)-365(edge)-364(set)-365(of)]TJ/F22 10.9091 Tf 184.477 0 Td [(G)]TJ/F15 10.9091 Tf 8.577 0 Td [(,)]TJ -409.421 -13.549 Td [(resp)-28(ectiv)28(ely)83(.)-466(Tw)28(o)-341(v)28(ertices)-341(are)-341(called)-340(adjacen)27(t)-340(if)-341(there)-341(is)-340(an)-341(edge)-341(connecting)-340(them.)-467(F)84(or)]TJ 0 -13.55 Td [(an)28(y)-238(in)27(teger)]TJ/F22 10.9091 Tf 54.621 0 Td [(\016)-316(>)]TJ/F15 10.9091 Tf 19.807 0 Td [(0,)-257(a)]TJ/F22 10.9091 Tf 19.344 0 Td [(\016)]TJ/F15 10.9091 Tf 5.261 0 Td [(-o)28(v)28(erlap)-239(part)1(ition)-239(of)]TJ/F22 10.9091 Tf 96.341 0 Td [(W)]TJ/F15 10.9091 Tf 14.416 0 Td [(can)-238(b)-28(e)-238(de\014ned)-238(recursiv)28(e)-1(l)1(y)-239(as)-238(follo)28(ws.)-413(Giv)28(en)]TJ -209.79 -13.549 Td [(a)-344(0-o)28(v)28(erlap)-344(\050or)-343(non-o)28(v)27(erlappin)1(g\051)-344(partition)-344(of)]TJ/F22 10.9091 Tf 216.25 0 Td [(W)]TJ/F15 10.9091 Tf 11.818 0 Td [(,)-346(i.e.)-344(a)-343(set)-344(of)]TJ/F22 10.9091 Tf 63.377 0 Td [(m)]TJ/F15 10.9091 Tf 13.327 0 Td [(disjoin)28(t)-344(nonempt)28(y)-343(s)-1(ets)]TJ/F22 10.9091 Tf -304.772 -13.549 Td [(W)]TJ/F20 7.9701 Tf 11.818 3.959 Td [(0)]TJ/F23 7.9701 Tf -1.515 -7.015 Td [(i)]TJ/F25 10.9091 Tf 9.278 3.056 Td [(\032)]TJ/F22 10.9091 Tf 11.515 0 Td [(W)]TJ/F15 10.9091 Tf 14.857 0 Td [(suc)28(h)-279(that)]TJ/F25 10.9091 Tf 47.047 0 Td [([)]TJ/F23 7.9701 Tf 7.273 3.959 Td [(m)]TJ 0 -7.015 Td [(i)]TJ/F20 7.9701 Tf 2.883 0 Td [(=1)]TJ/F22 10.9091 Tf 11.319 3.056 Td [(W)]TJ/F20 7.9701 Tf 11.818 3.959 Td [(0)]TJ/F23 7.9701 Tf -1.515 -7.015 Td [(i)]TJ/F15 10.9091 Tf 9.278 3.056 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(W)]TJ/F15 10.9091 Tf 11.818 0 Td [(,)-289(a)]TJ/F22 10.9091 Tf 14.682 0 Td [(\016)]TJ/F15 10.9091 Tf 5.261 0 Td [(-o)28(v)28(erlap)-279(partition)-278(of)]TJ/F22 10.9091 Tf 97.662 0 Td [(W)]TJ/F15 10.9091 Tf 14.856 0 Td [(is)-279(obtain)1(e)-1(d)-278(b)28(y)-279(considering)]TJ -289.85 -13.549 Td [(the)-307(sets)]TJ/F22 10.9091 Tf 39.538 0 Td [(W)]TJ/F23 7.9701 Tf 11.818 3.959 Td [(\016)]TJ -1.515 -7.015 Td [(i)]TJ/F25 10.9091 Tf 9.09 3.056 Td [(\033)]TJ/F22 10.9091 Tf 11.515 0 Td [(W)]TJ/F23 7.9701 Tf 11.818 4.588 Td [(\016)]TJ/F26 7.9701 Tf 4.046 0 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F23 7.9701 Tf -12.148 -7.845 Td [(i)]TJ/F15 10.9091 Tf 20.225 3.257 Td [(obtained)-306(b)27(y)-306(including)-307(the)-306(v)28(ertices)-307(that)-307(are)-306(adjacen)28(t)-307(to)-307(an)28(y)-306(v)27(ertex)]TJ -100.974 -13.549 Td [(in)]TJ/F22 10.9091 Tf 12.728 0 Td [(W)]TJ/F23 7.9701 Tf 11.818 4.587 Td [(\016)]TJ/F26 7.9701 Tf 4.046 0 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F23 7.9701 Tf -12.147 -7.844 Td [(i)]TJ/F15 10.9091 Tf 16.88 3.257 Td [(.)]TJ -22.974 -15.27 Td [(Let)]TJ/F22 10.9091 Tf 18.79 0 Td [(n)]TJ/F23 7.9701 Tf 6.548 3.959 Td [(\016)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf 7.425 3.055 Td [(b)-28(e)-264(the)-264(size)-264(of)]TJ/F22 10.9091 Tf 63.705 0 Td [(W)]TJ/F23 7.9701 Tf 11.818 3.959 Td [(\016)]TJ -1.515 -7.014 Td [(i)]TJ/F15 10.9091 Tf 8.94 3.055 Td [(and)]TJ/F22 10.9091 Tf 20.457 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.959 Td [(\016)]TJ -0.084 -7.014 Td [(i)]TJ/F25 10.9091 Tf 7.659 3.055 Td [(2)-278(<)]TJ/F23 7.9701 Tf 18.182 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.138 2.813 Td [(\016)]TJ 0 -5.395 Td [(i)]TJ/F26 7.9701 Tf 4.089 2.582 Td [(\002)]TJ/F23 7.9701 Tf 6.587 0 Td [(n)]TJ/F15 10.9091 Tf 8.517 -3.959 Td [(the)-264(restriction)-264(op)-28(erator)-264(that)-264(maps)-264(a)-264(v)28(ector)]TJ/F22 10.9091 Tf -211.56 -14.974 Td [(v)]TJ/F25 10.9091 Tf 8.993 0 Td [(2)-304(<)]TJ/F23 7.9701 Tf 18.465 3.959 Td [(n)]TJ/F15 10.9091 Tf 9.443 -3.959 Td [(on)28(to)-349(the)-349(v)28(ector)]TJ/F22 10.9091 Tf 76.601 0 Td [(v)]TJ/F23 7.9701 Tf 5.679 3.959 Td [(\016)]TJ -0.391 -7.015 Td [(i)]TJ/F25 10.9091 Tf 8.249 3.056 Td [(2)-304(<)]TJ/F23 7.9701 Tf 18.465 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.138 2.812 Td [(\016)]TJ 0 -5.395 Td [(i)]TJ/F15 10.9091 Tf 8.394 -1.376 Td [(con)28(taining)-349(the)-349(comp)-28(onen)28(ts)-349(of)]TJ/F22 10.9091 Tf 144.984 0 Td [(v)]TJ/F15 10.9091 Tf 9.485 0 Td [(corresp)-28(onding)-349(to)-348(the)]TJ -313.505 -14.975 Td [(v)28(ertices)-369(in)]TJ/F22 10.9091 Tf 52.99 0 Td [(W)]TJ/F23 7.9701 Tf 11.819 3.959 Td [(\016)]TJ -1.516 -7.014 Td [(i)]TJ/F15 10.9091 Tf 6.06 3.055 Td [(.)-551(The)-369(transp)-28(ose)-369(of)]TJ/F22 10.9091 Tf 94 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 3.959 Td [(\016)]TJ -0.085 -7.014 Td [(i)]TJ/F15 10.9091 Tf 8.654 3.055 Td [(is)-369(a)-369(prolongation)-369(op)-28(erator)-368(from)]TJ/F25 10.9091 Tf 155.46 0 Td [(<)]TJ/F23 7.9701 Tf 7.879 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.138 2.813 Td [(\016)]TJ 0 -5.395 Td [(i)]TJ/F15 10.9091 Tf 8.613 -1.377 Td [(to)]TJ/F25 10.9091 Tf 13.723 0 Td [(<)]TJ/F23 7.9701 Tf 7.878 3.959 Td [(n)]TJ/F15 10.9091 Tf 5.637 -3.959 Td [(.)-551(The)]TJ -384.618 -14.974 Td [(matrix)]TJ/F22 10.9091 Tf 35.134 0 Td [(A)]TJ/F23 7.9701 Tf 8.181 3.959 Td [(\016)]TJ 0 -7.015 Td [(i)]TJ/F15 10.9091 Tf 7.575 3.056 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.959 Td [(\016)]TJ -0.084 -7.015 Td [(i)]TJ/F22 10.9091 Tf 4.629 3.056 Td [(A)]TJ/F15 10.9091 Tf 8.181 0 Td [(\050)]TJ/F22 10.9091 Tf 4.243 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.959 Td [(\016)]TJ -0.084 -7.015 Td [(i)]TJ/F15 10.9091 Tf 4.628 3.056 Td [(\051)]TJ/F23 7.9701 Tf 4.243 3.959 Td [(T)]TJ/F25 10.9091 Tf 9.634 -3.959 Td [(2)-278(<)]TJ/F23 7.9701 Tf 18.182 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.138 2.812 Td [(\016)]TJ 0 -5.395 Td [(i)]TJ/F26 7.9701 Tf 4.09 2.583 Td [(\002)]TJ/F23 7.9701 Tf 6.587 0 Td [(n)]TJ/F24 5.9776 Tf 5.138 2.812 Td [(\016)]TJ 0 -5.395 Td [(i)]TJ/F15 10.9091 Tf 7.872 -1.376 Td [(can)-301(b)-28(e)-301(considered)-301(as)-301(a)-301(restriction)-301(of)]TJ/F22 10.9091 Tf 172.964 0 Td [(A)]TJ/F15 10.9091 Tf 11.466 0 Td [(corresp)-28(onding)]TJ -345.966 -13.549 Td [(to)-333(the)-334(set)]TJ/F22 10.9091 Tf 49.152 0 Td [(W)]TJ/F23 7.9701 Tf 11.818 3.958 Td [(\016)]TJ -1.515 -7.014 Td [(i)]TJ/F15 10.9091 Tf 6.059 3.056 Td [(.)]TJ -48.577 -13.643 Td [(The)]TJ/F18 10.9091 Tf 22.424 0 Td [(classic)51(al)-358(one-l)1(evel)-358(AS)]TJ/F15 10.9091 Tf 106.723 0 Td [(preconditioner)-333(is)-334(de\014n)1(e)-1(d)-333(b)28(y)]TJ/F22 10.9091 Tf -4.792 -32.517 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.022 Td [(AS)]TJ/F15 10.9091 Tf 15.538 3.434 Td [(=)]TJ/F23 7.9701 Tf 15.649 13.637 Td [(m)]TJ/F28 10.9091 Tf -4.134 -3.273 Td [(X)]TJ/F23 7.9701 Tf 1.027 -23.451 Td [(i)]TJ/F20 7.9701 Tf 2.883 0 Td [(=1)]TJ/F15 10.9091 Tf 11.848 13.087 Td [(\050)]TJ/F22 10.9091 Tf 4.242 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 4.505 Td [(\016)]TJ -0.085 -7.201 Td [(i)]TJ/F15 10.9091 Tf 4.629 2.696 Td [(\051)]TJ/F23 7.9701 Tf 4.242 4.505 Td [(T)]TJ/F15 10.9091 Tf 6.605 -4.505 Td [(\050)]TJ/F22 10.9091 Tf 4.242 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 4.505 Td [(\016)]TJ 0 -7.201 Td [(i)]TJ/F15 10.9091 Tf 4.544 2.696 Td [(\051)]TJ/F26 7.9701 Tf 4.243 4.505 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 4.732 -4.505 Td [(R)]TJ/F23 7.9701 Tf 8.368 4.505 Td [(\016)]TJ -0.084 -7.201 Td [(i)]TJ/F22 10.9091 Tf 4.628 2.696 Td [(;)]TJ/F15 10.9091 Tf -268.129 -35.907 Td [(where)]TJ/F22 10.9091 Tf 32.336 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.958 Td [(\016)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf 8.972 3.056 Td [(is)-406(assumed)-406(to)-406(b)-27(e)-406(nonsingular.)-662(Its)-406(application)-405(to)-406(a)-406(v)28(ector)]TJ/F22 10.9091 Tf 283.186 0 Td [(v)]TJ/F25 10.9091 Tf 10.027 0 Td [(2)-399(<)]TJ/F23 7.9701 Tf 19.5 3.958 Td [(n)]TJ/F15 10.9091 Tf 10.064 -3.958 Td [(within)-406(a)]TJ -372.267 -13.55 Td [(Krylo)28(v)-333(s)-1(ol)1(v)27(er)-333(requires)-333(the)-334(follo)28(wing)-333(three)-333(ste)-1(p)1(s)-1(:)]TJ -0 g 0 G - 13.334 -22.89 Td [(1.)]TJ -0 g 0 G - [-500(restriction)-333(of)]TJ/F22 10.9091 Tf 78.606 0 Td [(v)]TJ/F15 10.9091 Tf 9.315 0 Td [(as)]TJ/F22 10.9091 Tf 13.394 0 Td [(v)]TJ/F23 7.9701 Tf 5.288 -1.636 Td [(i)]TJ/F15 10.9091 Tf 6.412 1.636 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.959 Td [(\016)]TJ -0.084 -7.014 Td [(i)]TJ/F22 10.9091 Tf 4.628 3.055 Td [(v)]TJ/F15 10.9091 Tf 5.68 0 Td [(,)]TJ/F22 10.9091 Tf 6.666 0 Td [(i)]TJ/F15 10.9091 Tf 6.789 0 Td [(=)-278(1)]TJ/F22 10.9091 Tf 16.97 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(m)]TJ/F15 10.9091 Tf 33.82 0 Td [(;)]TJ + [(-)]TJ +0 0 1 rg 0 0 1 RG + [(4.3)]TJ 0 g 0 G - -207.366 -22.89 Td [(2.)]TJ + [(.)-444(F)83(or)]TJ 0 -13.549 Td [(further)-333(details)-334(th)1(e)-334(reader)-333(is)-334(r)1(e)-1(f)1(e)-1(r)1(re)-1(d)-333(to)-333([)]TJ +1 0 0 rg 1 0 0 RG + [(3)]TJ 0 g 0 G - [-500(solution)-333(of)-333(the)-334(linear)-333(systems)]TJ/F22 10.9091 Tf 157.242 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(\016)]TJ 0 -7.015 Td [(i)]TJ/F22 10.9091 Tf 4.544 3.056 Td [(w)]TJ/F23 7.9701 Tf 7.81 -1.637 Td [(i)]TJ/F15 10.9091 Tf 6.412 1.637 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(v)]TJ/F23 7.9701 Tf 5.288 -1.637 Td [(i)]TJ/F15 10.9091 Tf 3.381 1.637 Td [(,)]TJ/F22 10.9091 Tf 6.667 0 Td [(i)]TJ/F15 10.9091 Tf 6.788 0 Td [(=)-278(1)]TJ/F22 10.9091 Tf 16.97 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(m)]TJ/F15 10.9091 Tf 33.821 0 Td [(;)]TJ + [(,)]TJ +1 0 0 rg 1 0 0 RG + [-333(4)]TJ 0 g 0 G - -268.62 -22.89 Td [(3.)]TJ + [(,)]TJ +1 0 0 rg 1 0 0 RG + [-334(5)]TJ 0 g 0 G - [-500(prolongation)-333(and)-333(sum)-334(of)-333(the)]TJ/F22 10.9091 Tf 153.121 0 Td [(w)]TJ/F23 7.9701 Tf 7.81 -1.637 Td [(i)]TJ/F15 10.9091 Tf 3.381 1.637 Td [('s,)-333(i.e.)]TJ/F22 10.9091 Tf 32.788 0 Td [(w)]TJ/F15 10.9091 Tf 11.134 0 Td [(=)]TJ/F28 10.9091 Tf 11.515 8.181 Td [(P)]TJ/F23 7.9701 Tf 11.515 -3.154 Td [(m)]TJ 0 -8.253 Td [(i)]TJ/F20 7.9701 Tf 2.883 0 Td [(=1)]TJ/F15 10.9091 Tf 11.319 3.226 Td [(\050)]TJ/F22 10.9091 Tf 4.243 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.958 Td [(\016)]TJ -0.084 -7.014 Td [(i)]TJ/F15 10.9091 Tf 4.628 3.056 Td [(\051)]TJ/F23 7.9701 Tf 4.243 3.958 Td [(T)]TJ/F22 10.9091 Tf 6.604 -3.958 Td [(w)]TJ/F23 7.9701 Tf 7.81 -1.637 Td [(i)]TJ/F15 10.9091 Tf 3.381 1.637 Td [(.)]TJ -297.992 -22.891 Td [(Note)-473(that)-472(the)-473(l)1(inear)-473(systems)-473(at)-472(step)-473(2)-472(are)-473(usually)-472(solv)28(ed)-473(appro)28(ximately)83(,)-507(e.g.)-472(using)]TJ 0 -13.549 Td [(incomplete)-333(LU)-334(factorizations)-333(suc)28(h)-334(as)-333(ILU\050)]TJ/F22 10.9091 Tf 202.637 0 Td [(p)]TJ/F15 10.9091 Tf 5.488 0 Td [(\051,)-333(MILU\050)]TJ/F22 10.9091 Tf 44.091 0 Td [(p)]TJ/F15 10.9091 Tf 5.489 0 Td [(\051)-333(and)-334(ILU\050)]TJ/F22 10.9091 Tf 52.273 0 Td [(p;)-167(t)]TJ/F15 10.9091 Tf 14.276 0 Td [(\051)-333([)]TJ + [(,)]TJ 1 0 0 rg 1 0 0 RG - [(22)]TJ + [-333(9)]TJ 0 g 0 G - [(,)-334(Chapter)-333(10].)]TJ + [(].)]TJ 16.937 -14.105 Td [(W)83(e)-430(note)-430(that)-430(opti)1(m)-1(al)-429(m)27(ultigrid)-430(pr)1(e)-1(cond)1(itioners)-430(do)-430(not)-430(necessarily)-430(corresp)-28(ond)-430(to)]TJ -16.937 -13.549 Td [(minim)28(um)-486(execution)-485(times)-485(in)-485(a)-486(parallel)-485(setting.)-900(Indeed,)-523(to)-486(obtain)-485(e\013ectiv)28(e)-486(parallel)]TJ 0 -13.549 Td [(m)28(ultigrid)-403(preconditioners,)-421(a)-403(tradeo\013)-403(b)-28(et)28(w)28(een)-404(the)-403(optimalit)28(y)-403(and)-403(the)-403(cost)-404(of)-403(buildi)1(ng)]TJ 0 -13.549 Td [(and)-223(applying)-223(the)-223(smo)-28(others)-223(and)-223(the)-223(coarse-space)-223(corrections)-223(m)27(ust)-223(b)-27(e)-224(ac)28(hiev)28(ed.)-408(E\013ectiv)28(e)]TJ 0 -13.55 Td [(parallel)-324(preconditioners)-324(re)-1(q)1(uire)-325(algorithmic)-324(scalabilit)28(y)-325(to)-324(b)-28(e)-324(coupled)-324(with)-325(implemen)28(ta-)]TJ 0 -13.549 Td 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rg 1 0 0 RG - [(5)]TJ + [(24)]TJ 0 g 0 G - [(,)]TJ + [(].)-446(An)-333(example)-334(of)-333(s)-1(u)1(c)27(h)-333(a)-334(com)28(bination,)]TJ 0 -13.549 Td [(kno)28(wn)-448(as)-448(V-cycle,)-476(is)-448(giv)28(en)-448(in)-447(Figure)]TJ +0 0 1 rg 0 0 1 RG + [-448(1)]TJ +0 g 0 G + [(.)-788(In)-447(this)-448(case,)-476(a)-448(single)-448(iteration)-448(of)-447(the)-448(same)]TJ 0 -13.549 Td [(smo)-28(other)-333(is)-332(used)-333(b)-28(efore)-333(and)-332(after)-333(the)-333(the)-333(recursiv)28(e)-333(call)-333(to)-332(the)-333(V-cycle)-333(\050i.e.,)-333(in)-333(th)1(e)-333(pre-)]TJ 0 -13.55 Td [(smo)-28(othing)-409(an)1(d)-409(p)-28(ost-smo)-28(othing)-409(ph)1(as)-1(es\051;)-446(ho)28(w)27(ev)28(er,)-427(di\013eren)27(t)-408(c)27(hoices)-409(can)-408(b)-28(e)-409(p)-28(erformed.)]TJ 0 -13.549 Td [(Other)-405(cycles)-405(c)-1(an)-405(b)-27(e)-406(de\014ned;)-441(in)-405(MLD2P4,)-423(w)28(e)-405(implemen)27(ted)-405(the)-405(standard)-405(V-cycle)-405(and)]TJ 0 -13.549 Td [(W-cycle)-333([)]TJ +1 0 0 rg 1 0 0 RG + [(2)]TJ +0 g 0 G + [(],)-334(and)-333(a)-333(v)28(e)-1(r)1(s)-1(ion)-333(of)-333(the)-333(K-cyc)-1(l)1(e)-334(describ)-28(ed)-333(in)-333([)]TJ 1 0 0 rg 1 0 0 RG - [-387(15)]TJ + [(23)]TJ 0 g 0 G - [(].)-605(It)-387(is)-386(obtained)]TJ -235.209 -13.549 Td [(b)28(y)-408(zeroing)-407(the)-408(comp)-28(onen)28(ts)-407(of)]TJ/F22 10.9091 Tf 148.078 0 Td [(w)]TJ/F23 7.9701 Tf 7.81 -1.637 Td [(i)]TJ/F15 10.9091 Tf 7.827 1.637 Td [(corresp)-28(onding)-407(to)-408(the)-407(o)28(v)27(erlappin)1(g)-408(v)28(ertices)-408(when)-407(ap-)]TJ -163.715 -13.549 Td [(plying)-349(the)-348(prolongation.)-491(Therefore,)-353(RAS)-349(di\013ers)-348(from)-349(classical)-349(AS)-349(b)28(y)-349(the)-349(prolongation)]TJ 0 -13.55 Td [(op)-28(erators,)-400(whic)27(h)-387(are)-387(substituted)-387(b)28(y)-387(\050)]TJ 182.691 2.758 Td [(~)]TJ/F22 10.9091 Tf -2.365 -2.758 Td [(R)]TJ/F20 7.9701 Tf 8.367 3.959 Td [(0)]TJ/F23 7.9701 Tf -0.084 -7.014 Td [(i)]TJ/F15 10.9091 Tf 4.816 3.055 Td [(\051)]TJ/F23 7.9701 Tf 4.243 3.959 Td [(T)]TJ/F25 10.9091 Tf 10.613 -3.959 Td [(2)-368(<)]TJ/F23 7.9701 Tf 19.161 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.138 2.813 Td [(\016)]TJ 0 -5.395 Td [(i)]TJ/F26 7.9701 Tf 4.09 2.582 Td [(\002)]TJ/F23 7.9701 Tf 6.586 0 Td [(n)]TJ/F15 10.9091 Tf 5.637 -3.959 Td [(,)-401(where)]TJ 41.899 2.758 Td [(~)]TJ/F22 10.9091 Tf -2.366 -2.758 Td [(R)]TJ/F20 7.9701 Tf 8.368 3.959 Td [(0)]TJ/F23 7.9701 Tf -0.085 -7.014 Td [(i)]TJ/F15 10.9091 Tf 9.041 3.055 Td [(is)-387(obtained)-387(b)28(y)-388(zeroing)]TJ -305.75 -13.549 Td [(the)-333(ro)28(w)-1(s)-333(of)]TJ/F22 10.9091 Tf 56.454 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 3.959 Td [(\016)]TJ -0.084 -7.014 Td [(i)]TJ/F15 10.9091 Tf 8.264 3.055 Td [(corresp)-28(onding)-333(to)-333(the)-334(v)28(ertices)-333(in)]TJ/F22 10.9091 Tf 154.455 0 Td [(W)]TJ/F23 7.9701 Tf 11.818 3.959 Td [(\016)]TJ -1.515 -7.014 Td [(i)]TJ/F25 10.9091 Tf 6.059 3.055 Td [(n)]TJ/F22 10.9091 Tf 5.455 0 Td [(W)]TJ/F20 7.9701 Tf 11.818 3.959 Td [(0)]TJ/F23 7.9701 Tf -1.515 -7.014 Td [(i)]TJ/F15 10.9091 Tf 6.247 3.055 Td [(:)]TJ/F22 10.9091 Tf -127.804 -29.635 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.022 Td [(R)-6(AS)]TJ/F15 10.9091 Tf 21.895 3.434 Td [(=)]TJ/F23 7.9701 Tf 15.648 13.637 Td [(m)]TJ/F28 10.9091 Tf -4.133 -3.273 Td [(X)]TJ/F23 7.9701 Tf 1.027 -23.451 Td [(i)]TJ/F20 7.9701 Tf 2.883 0 Td [(=1)]TJ/F15 10.9091 Tf 11.847 13.087 Td [(\050)]TJ 6.608 2.758 Td [(~)]TJ/F22 10.9091 Tf -2.365 -2.758 Td [(R)]TJ/F20 7.9701 Tf 8.367 4.505 Td [(0)]TJ/F23 7.9701 Tf -0.084 -7.201 Td [(i)]TJ/F15 10.9091 Tf 4.817 2.696 Td [(\051)]TJ/F23 7.9701 Tf 4.242 4.505 Td [(T)]TJ/F15 10.9091 Tf 6.605 -4.505 Td [(\050)]TJ/F22 10.9091 Tf 4.242 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 4.505 Td [(\016)]TJ 0 -7.201 Td [(i)]TJ/F15 10.9091 Tf 4.544 2.696 Td [(\051)]TJ/F26 7.9701 Tf 4.242 4.505 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F22 10.9091 Tf 4.732 -4.505 Td [(R)]TJ/F23 7.9701 Tf 8.368 4.505 Td [(\016)]TJ -0.085 -7.201 Td [(i)]TJ/F22 10.9091 Tf 4.629 2.696 Td [(:)]TJ/F15 10.9091 Tf -271.401 -30.78 Td [(Analogously)83(,)-333(the)-333(AS)-333(v)55(arian)28(t)-333(called)]TJ/F18 10.9091 Tf 168.576 0 Td [(AS)-358(with)-358(Harmonic)-357(extension)-358(\050ASH\051)]TJ/F15 10.9091 Tf 176.608 0 Td [(is)-333(de\014ned)-334(b)28(y)]TJ/F22 10.9091 Tf -207.736 -29.306 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.022 Td [(AS)-56(H)]TJ/F15 10.9091 Tf 23.037 3.434 Td [(=)]TJ/F23 7.9701 Tf 15.649 13.636 Td [(m)]TJ/F28 10.9091 Tf -4.134 -3.273 Td [(X)]TJ/F23 7.9701 Tf 1.027 -23.451 Td [(i)]TJ/F20 7.9701 Tf 2.883 0 Td [(=1)]TJ/F15 10.9091 Tf 11.848 13.088 Td [(\050)]TJ/F22 10.9091 Tf 4.242 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 4.504 Td [(\016)]TJ -0.085 -7.201 Td [(i)]TJ/F15 10.9091 Tf 4.629 2.697 Td [(\051)]TJ/F23 7.9701 Tf 4.242 4.504 Td [(T)]TJ/F15 10.9091 Tf 6.605 -4.504 Td [(\050)]TJ/F22 10.9091 Tf 4.242 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 4.504 Td [(\016)]TJ 0 -7.201 Td [(i)]TJ/F15 10.9091 Tf 4.544 2.697 Td [(\051)]TJ/F26 7.9701 Tf 4.243 4.504 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F15 10.9091 Tf 7.098 -1.747 Td [(~)]TJ/F22 10.9091 Tf -2.366 -2.757 Td [(R)]TJ/F20 7.9701 Tf 8.368 4.504 Td [(0)]TJ/F23 7.9701 Tf -0.084 -7.201 Td [(i)]TJ/F22 10.9091 Tf 4.816 2.697 Td [(:)]TJ/F15 10.9091 Tf -271.972 -30.174 Td [(W)83(e)-334(note)-335(that)-335(for)]TJ/F22 10.9091 Tf 83.422 0 Td [(\016)]TJ/F15 10.9091 Tf 8.315 0 Td [(=)-280(0)-335(the)-334(three)-335(v)56(arian)27(ts)-334(of)-335(the)-335(AS)-334(preconditioner)-335(are)-334(all)-335(equal)-335(to)-334(the)]TJ -91.737 -13.55 Td [(blo)-28(c)28(k-Jacobi)-333(preconditioner.)]TJ 16.936 -13.549 Td [(As)-358(already)-359(ob)1(s)-1(erv)28(ed,)-364(the)-358(c)-1(on)28(v)28(ergence)-358(rate)-359(of)-358(the)-358(one-lev)28(el)-359(S)1(c)27(h)28(w)28(arz)-358(preconditioned)]TJ -16.936 -13.549 Td [(iterativ)28(e)-358(s)-1(olv)28(ers)-358(deteriorates)-358(as)-359(the)-358(n)28(um)28(b)-28(er)]TJ/F22 10.9091 Tf 211.422 0 Td [(m)]TJ/F15 10.9091 Tf 13.487 0 Td [(of)-358(partitions)-358(of)]TJ/F22 10.9091 Tf 75.453 0 Td [(W)]TJ/F15 10.9091 Tf 15.727 0 Td [(increases)-358([)]TJ + [(].)]TJ/F17 11.9552 Tf 0 -29.053 Td [(4.2)-1125(Smo)-31(othed)-375(Aggregation)]TJ/F15 10.9091 Tf 0 -20.595 Td [(In)-374(order)-374(to)-374(de\014ne)-375(t)1(he)-375(prolongator)]TJ/F22 10.9091 Tf 167.339 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(,)-384(used)-375(to)-374(compute)-374(the)-374(coarse-lev)28(el)-375(matrix)]TJ/F22 10.9091 Tf 204.32 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F15 10.9091 Tf 11.319 -3.959 Td [(,)]TJ -409.421 -13.549 Td [(MLD2P4)-319(uses)-320(the)-319(smo)-28(othed)-319(aggregation)-319(algorithm)-320(describ)-27(ed)-320(in)-319([)]TJ 1 0 0 rg 1 0 0 RG - [(7)]TJ + [(1)]TJ 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-359(23)]TJ + [-319(29)]TJ 0 g 0 G - [(].)-519(T)83(o)]TJ -316.089 -13.549 Td [(reduce)-308(the)-308(dep)-28(endency)-307(of)-308(the)-308(n)28(um)27(b)-27(er)-308(of)-308(iterations)-308(on)-308(the)-308(degree)-308(of)-308(parall)1(e)-1(li)1(s)-1(m)-307(w)27(e)-308(ma)28(y)]TJ 0 -13.549 Td [(in)28(tro)-28(duce)-275(a)-276(global)-275(coupling)-276(among)-275(the)-275(o)27(v)28(erlapping)-275(partitions)-276(b)28(y)-275(de\014ning)-276(a)-275(coarse-space)]TJ 0 -13.55 Td [(appro)28(ximation)]TJ/F22 10.9091 Tf 72.12 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 -1.688 Td [(C)]TJ/F15 10.9091 Tf 10.095 1.688 Td [(of)-275(the)-275(matrix)]TJ/F22 10.9091 Tf 64.783 0 Td [(A)]TJ/F15 10.9091 Tf 8.182 0 Td [(.)-425(In)-275(a)-275(pur)1(e)-275(algebraic)-275(setting,)]TJ/F22 10.9091 Tf 137.15 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 -1.688 Td [(C)]TJ/F15 10.9091 Tf 10.095 1.688 Td [(is)-275(usually)-275(buil)1(t)-275(with)]TJ -318.789 -13.549 Td [(the)-321(Galerkin)-322(appr)1(oac)27(h.)-440(Giv)28(en)-322(a)-321(set)]TJ/F22 10.9091 Tf 171.697 0 Td [(W)]TJ/F23 7.9701 Tf 10.303 -1.689 Td [(C)]TJ/F15 10.9091 Tf 10.602 1.689 Td [(of)]TJ/F18 10.9091 Tf 12.294 0 Td [(c)51(o)51(arse)-346(vertic)51(es)]TJ/F15 10.9091 Tf 68.461 0 Td [(,)-324(with)-321(size)]TJ/F22 10.9091 Tf 51.815 0 Td [(n)]TJ/F23 7.9701 Tf 6.548 -1.689 Td [(C)]TJ/F15 10.9091 Tf 7.097 1.689 Td [(,)-324(and)-321(a)-321(suitable)]TJ -338.817 -13.549 Td [(restriction)-333(op)-28(erator)]TJ/F22 10.9091 Tf 96.243 0 Td [(R)]TJ/F23 7.9701 Tf 8.283 -1.689 Td [(C)]TJ/F25 10.9091 Tf 10.127 1.689 Td [(2)-278(<)]TJ/F23 7.9701 Tf 18.182 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.34 Td [(C)]TJ/F26 7.9701 Tf 6.194 1.34 Td [(\002)]TJ/F23 7.9701 Tf 6.587 0 Td [(n)]TJ/F15 10.9091 Tf 5.636 -3.959 Td [(,)]TJ/F22 10.9091 Tf 6.667 0 Td [(A)]TJ/F23 7.9701 Tf 8.181 -1.689 Td [(C)]TJ/F15 10.9091 Tf 10.734 1.689 Td [(is)-333(de\014ned)-334(as)]TJ/F22 10.9091 Tf -10.129 -20.973 Td [(A)]TJ/F23 7.9701 Tf 8.182 -1.689 Td [(C)]TJ/F15 10.9091 Tf 10.127 1.689 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(R)]TJ/F23 7.9701 Tf 8.283 -1.689 Td [(C)]TJ/F22 10.9091 Tf 7.097 1.689 Td [(AR)]TJ/F23 7.9701 Tf 16.549 4.504 Td [(T)]TJ -0.084 -7.201 Td [(C)]TJ/F15 10.9091 Tf -233.512 -18.277 Td [(and)-496(the)-496(coars)-1(e-lev)28(el)-496(correction)-496(matrix)-497(to)-496(b)-28(e)-496(com)28(bined)-496(with)-497(a)-496(generic)-496(one-lev)28(el)-497(AS)]TJ 0 -13.549 Td [(preconditioner)]TJ/F22 10.9091 Tf 72.182 0 Td [(M)]TJ/F20 7.9701 Tf 10.583 -1.688 Td [(1)]TJ/F23 7.9701 Tf 4.235 0 Td [(L)]TJ/F15 10.9091 Tf 9.893 1.688 Td [(is)-333(obtained)-334(as)]TJ/F22 10.9091 Tf 63.869 -20.973 Td [(M)]TJ/F26 7.9701 Tf 11.772 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.022 Td [(C)]TJ/F15 10.9091 Tf 15.539 3.434 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 4.504 Td [(T)]TJ -0.084 -7.201 Td [(C)]TJ/F22 10.9091 Tf 7.097 2.697 Td [(A)]TJ/F26 7.9701 Tf 8.181 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F23 7.9701 Tf -6.587 -8.022 Td [(C)]TJ/F22 10.9091 Tf 11.319 3.434 Td [(R)]TJ/F23 7.9701 Tf 8.283 -1.689 Td [(C)]TJ/F22 10.9091 Tf 7.097 1.689 Td [(;)]TJ/F15 10.9091 Tf -248.659 -20.973 Td [(where)]TJ/F22 10.9091 Tf 31.91 0 Td [(A)]TJ/F23 7.9701 Tf 8.181 -1.689 Td [(C)]TJ/F15 10.9091 Tf 11.098 1.689 Td [(is)-367(assumed)-366(to)-367(b)-28(e)-367(nonsingular)1(.)-545(The)-367(application)-366(of)]TJ/F22 10.9091 Tf 240.883 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.022 Td [(C)]TJ/F15 10.9091 Tf 16.509 3.434 Td [(to)-367(a)-366(v)27(ector)]TJ/F22 10.9091 Tf 56.274 0 Td [(v)]TJ/F15 10.9091 Tf 9.68 0 Td [(corre-)]TJ -385.118 -13.549 Td [(sp)-28(onds)-325(to)-325(a)-325(restriction,)-327(a)-325(solution)-325(and)-325(a)-325(prolongation)-325(step;)-328(the)-325(solution)-325(step,)-327(in)28(v)27(olv)1(ing)]TJ 0 -13.55 Td [(the)-333(matrix)]TJ/F22 10.9091 Tf 54.273 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 -1.688 Td [(C)]TJ/F15 10.9091 Tf 7.097 1.688 Td [(,)-333(ma)27(y)-333(b)-28(e)-333(carried)-333(out)-334(also)-333(appro)28(ximately)83(.)]TJ -52.615 -13.549 Td [(The)-309(com)28(bination)-309(of)]TJ/F22 10.9091 Tf 96.176 0 Td [(M)]TJ/F23 7.9701 Tf 10.583 -1.689 Td [(C)]TJ/F15 10.9091 Tf 10.469 1.689 Td [(and)]TJ/F22 10.9091 Tf 20.947 0 Td [(M)]TJ/F20 7.9701 Tf 10.584 -1.689 Td [(1)]TJ/F23 7.9701 Tf 4.234 0 Td [(L)]TJ/F15 10.9091 Tf 9.629 1.689 Td [(ma)28(y)-309(b)-28(e)-309(p)-28(erformed)-309(in)-309(either)-309(an)-309(additiv)28(e)-309(or)-309(a)-309(m)27(ul-)]TJ -179.559 -13.549 Td [(tiplicativ)28(e)-275(framew)28(ork.)-425(In)-275(the)-275(former)-275(c)-1(ase,)-286(the)]TJ/F18 10.9091 Tf 217.192 0 Td [(two-level)-304(additive)]TJ/F15 10.9091 Tf 85.198 0 Td [(Sc)28(h)28(w)27(arz)-275(preconditioner)]TJ -302.39 -13.549 Td [(is)-333(obtained:)]TJ/F22 10.9091 Tf 153.971 -13.549 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.587 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ -7.776 -8.022 Td [(2)]TJ/F23 7.9701 Tf 4.234 0 Td [(LA)]TJ/F15 10.9091 Tf 15.631 3.435 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.587 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.022 Td [(C)]TJ/F15 10.9091 Tf 14.933 3.435 Td [(+)]TJ/F22 10.9091 Tf 10.909 0 Td [(M)]TJ/F26 7.9701 Tf 11.772 4.587 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ -7.776 -8.022 Td [(1)]TJ/F23 7.9701 Tf 4.234 0 Td [(L)]TJ/F22 10.9091 Tf 8.274 3.435 Td [(:)]TJ/F15 10.9091 Tf -255.45 -18.258 Td [(Applying)]TJ/F22 10.9091 Tf 48.201 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ -7.776 -8.022 Td [(2)]TJ/F23 7.9701 Tf 4.234 0 Td [(L)]TJ/F26 7.9701 Tf 5.759 0 Td [(\000)]TJ/F23 7.9701 Tf 6.586 0 Td [(A)]TJ/F15 10.9091 Tf 11.406 3.434 Td [(to)-418(a)-419(v)28(ector)]TJ/F22 10.9091 Tf 57.967 0 Td [(v)]TJ/F15 10.9091 Tf 10.245 0 Td [(within)-418(a)-419(Krylo)28(v)-418(solv)28(e)-1(r)-418(corresp)-28(onds)-418(to)-419(app)1(lying)]TJ/F22 10.9091 Tf 234.378 0 Td [(M)]TJ/F26 7.9701 Tf 11.772 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.022 Td [(C)]TJ/F15 10.9091 Tf -399.943 -11.551 Td [(and)]TJ/F22 10.9091 Tf 21.212 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ -7.775 -8.022 Td [(1)]TJ/F23 7.9701 Tf 4.234 0 Td [(L)]TJ/F15 10.9091 Tf 11.91 3.434 Td [(to)]TJ/F22 10.9091 Tf 13.333 0 Td [(v)]TJ/F15 10.9091 Tf 9.316 0 Td [(indep)-28(enden)28(tly)-333(and)-333(then)-334(summing)-333(up)-333(the)-334(results.)]TJ -53.652 -13.549 Td [(In)-355(the)-355(m)28(ultiplicativ)28(e)-355(case,)-360(the)-355(com)28(bination)-355(can)-355(b)-28(e)-355(p)-27(erformed)-355(b)28(y)-355(\014rst)-355(applying)-355(the)]TJ -16.937 -13.549 Td [(smo)-28(other)]TJ/F22 10.9091 Tf 47.667 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ -7.776 -8.022 Td [(1)]TJ/F23 7.9701 Tf 4.234 0 Td [(L)]TJ/F15 10.9091 Tf 11.911 3.434 Td [(and)-333(then)-333(the)-334(coarse-lev)28(el)-334(correction)-333(op)-28(erator)]TJ/F22 10.9091 Tf 217.485 0 Td [(M)]TJ/F26 7.9701 Tf 11.772 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.022 Td [(C)]TJ/F15 10.9091 Tf 12.508 3.434 Td [(:)]TJ/F22 10.9091 Tf -164.465 -21.579 Td [(w)]TJ/F15 10.9091 Tf 11.133 0 Td [(=)]TJ/F22 10.9091 Tf 11.516 0 Td [(M)]TJ/F26 7.9701 Tf 11.772 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ -7.776 -8.022 Td [(1)]TJ/F23 7.9701 Tf 4.234 0 Td [(L)]TJ/F22 10.9091 Tf 8.274 3.434 Td [(v)-36(;)]TJ -45.74 -13.789 Td [(z)]TJ/F15 10.9091 Tf 8.583 0 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(w)]TJ/F15 10.9091 Tf 10.528 0 Td [(+)]TJ/F22 10.9091 Tf 10.909 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.022 Td [(C)]TJ/F15 10.9091 Tf 12.508 3.434 Td [(\050)]TJ/F22 10.9091 Tf 4.243 0 Td [(v)]TJ/F25 10.9091 Tf 8.103 0 Td [(\000)]TJ/F22 10.9091 Tf 10.909 0 Td [(Aw)]TJ/F15 10.9091 Tf 16.286 0 Td [(\051;)]TJ + [(].)-440(The)-319(basic)-320(idea)]TJ 0 -13.549 Td [(of)-270(this)-271(algorithm)-270(is)-271(to)-270(build)-270(a)-271(coarse)-270(se)-1(t)-270(of)-270(indices)-271(\012)]TJ/F23 7.9701 Tf 243.122 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.621 0 Td [(+1)]TJ/F15 10.9091 Tf 14.27 -3.959 Td [(b)28(y)-271(suitabl)1(y)-271(grouping)-270(the)-271(indices)]TJ -262.013 -13.549 Td [(of)-350(\012)]TJ/F23 7.9701 Tf 20.481 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.934 -3.959 Td [(in)28(to)-350(disjoin)28(t)-350(sub)1(s)-1(ets)-349(\050aggregates\051,)-354(and)-350(to)-349(de\014ne)-350(the)-349(coarse)-1(-to-\014n)1(e)-350(space)-350(transfer)]TJ 0 g 0 G 0 g 0 G ET endstream endobj -338 0 obj +342 0 obj << -/Length 11371 +/Length 10301 >> stream 0 g 0 G @@ -1935,120 +1933,121 @@ stream BT /F41 10.9091 Tf 93.6 740.002 Td [(4)]TJ 0 g 0 G - [-378(Mul)67(ti-level)-378(Domain)-378(Decompo)1(siti)-1(o)1(n)-378(Ba)22(ck)22(gr)22(ound)]TJ/F15 10.9091 Tf 401.542 0 Td [(13)]TJ + [-378(Mul)67(tigrid)-378(Ba)22(ck)22(gr)23(ound)]TJ/F15 10.9091 Tf 401.542 0 Td [(13)]TJ 0 g 0 G - -401.542 -35.866 Td [(this)-278(corresp)-28(onds)-278(to)-278(the)-278(follo)28(wing)]TJ/F18 10.9091 Tf 156.322 0 Td [(two-level)-307(hybrid)-307(pr)51(e-smo)51(othe)52(d)]TJ/F15 10.9091 Tf 143.004 0 Td [(Sc)28(h)28(w)27(arz)-278(preconditioner:)]TJ/F22 10.9091 Tf -190.113 -25.536 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.587 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ -7.775 -8.022 Td [(2)]TJ/F23 7.9701 Tf 4.234 0 Td [(LH)]TJ/F26 7.9701 Tf 13.329 0 Td [(\000)]TJ/F23 7.9701 Tf 6.587 0 Td [(P)-148(R)-6(E)]TJ/F15 10.9091 Tf 23.153 3.435 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.587 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.022 Td [(C)]TJ/F15 10.9091 Tf 14.932 3.435 Td [(+)]TJ/F28 10.9091 Tf 10.909 8.836 Td [(\000)]TJ/F22 10.9091 Tf 5 -8.836 Td [(I)]TJ/F25 10.9091 Tf 8.076 0 Td [(\000)]TJ/F22 10.9091 Tf 10.909 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.587 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.021 Td [(C)]TJ/F22 10.9091 Tf 12.508 3.434 Td [(A)]TJ/F28 10.9091 Tf 8.182 8.836 Td [(\001)]TJ/F22 10.9091 Tf 6.818 -8.836 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.587 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ -7.776 -8.021 Td [(1)]TJ/F23 7.9701 Tf 4.234 0 Td [(L)]TJ/F22 10.9091 Tf 8.274 3.434 Td [(:)]TJ/F15 10.9091 Tf -300.208 -25.537 Td [(On)-368(the)-369(other)-368(hand,)-378(b)28(y)-368(applying)-368(the)-369(smo)-28(other)-368(after)-369(the)-368(coarse-lev)27(el)-368(correction,)-377(i.e.)-369(b)28(y)]TJ 0 -13.549 Td [(computing)]TJ/F22 10.9091 Tf 150.506 -13.871 Td [(w)]TJ/F15 10.9091 Tf 11.134 0 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(M)]TJ/F26 7.9701 Tf 11.772 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.022 Td [(C)]TJ/F22 10.9091 Tf 12.508 3.434 Td [(v)-36(;)]TJ -45.74 -13.789 Td [(z)]TJ/F15 10.9091 Tf 8.583 0 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(w)]TJ/F15 10.9091 Tf 10.528 0 Td [(+)]TJ/F22 10.9091 Tf 10.909 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ -7.776 -8.022 Td [(1)]TJ/F23 7.9701 Tf 4.235 0 Td [(L)]TJ/F15 10.9091 Tf 8.274 3.434 Td [(\050)]TJ/F22 10.9091 Tf 4.242 0 Td [(v)]TJ/F25 10.9091 Tf 8.103 0 Td [(\000)]TJ/F22 10.9091 Tf 10.91 0 Td [(Aw)]TJ/F15 10.9091 Tf 16.285 0 Td [(\051)]TJ/F22 10.9091 Tf 4.242 0 Td [(;)]TJ/F15 10.9091 Tf -258.915 -20.312 Td [(the)]TJ/F18 10.9091 Tf 18.788 0 Td [(two-level)-358(hybrid)-357(p)51(ost-smo)51(othe)51(d)]TJ/F15 10.9091 Tf 148.757 0 Td [(Sc)28(h)28(w)27(arz)-333(preconditioner)-333(is)-333(obtained:)]TJ/F22 10.9091 Tf -60.973 -25.537 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ -7.776 -8.022 Td [(2)]TJ/F23 7.9701 Tf 4.234 0 Td [(LH)]TJ/F26 7.9701 Tf 13.33 0 Td [(\000)]TJ/F23 7.9701 Tf 6.586 0 Td [(P)-148(O)-29(S)-56(T)]TJ/F15 10.9091 Tf 28.436 3.434 Td [(=)]TJ/F22 10.9091 Tf 11.516 0 Td [(M)]TJ/F26 7.9701 Tf 11.772 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ -7.776 -8.022 Td [(1)]TJ/F23 7.9701 Tf 4.234 0 Td [(L)]TJ/F15 10.9091 Tf 10.698 3.434 Td [(+)]TJ/F28 10.9091 Tf 10.909 8.836 Td [(\000)]TJ/F22 10.9091 Tf 5 -8.836 Td [(I)]TJ/F25 10.9091 Tf 8.076 0 Td [(\000)]TJ/F22 10.9091 Tf 10.909 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ -7.776 -8.022 Td [(1)]TJ/F23 7.9701 Tf 4.235 0 Td [(L)]TJ/F22 10.9091 Tf 8.274 3.434 Td [(A)]TJ/F28 10.9091 Tf 8.181 8.836 Td [(\001)]TJ/F22 10.9091 Tf 6.819 -8.836 Td [(M)]TJ/F26 7.9701 Tf 11.772 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.022 Td [(C)]TJ/F22 10.9091 Tf 12.508 3.434 Td [(:)]TJ/F15 10.9091 Tf -302.849 -25.537 Td [(One)-222(more)-222(v)55(arian)28(t)-222(of)-222(t)27(w)28(o-lev)28(el)-222(h)27(yb)1(rid)-223(preconditi)1(oner)-223(is)-222(obtained)-222(b)28(y)-222(applying)-222(the)-223(smo)-28(oth)1(e)-1(r)]TJ 0 -13.549 Td [(b)-28(efore)-239(and)-239(after)-239(the)-240(coarse-lev)28(el)-239(correction.)-413(In)-239(this)-240(case,)-258(the)-239(preconditioner)-239(is)-239(symm)-1(etric)]TJ 0 -13.549 Td [(if)]TJ/F22 10.9091 Tf 10 0 Td [(A)]TJ/F15 10.9091 Tf 8.182 0 Td [(,)]TJ/F22 10.9091 Tf 6.667 0 Td [(M)]TJ/F20 7.9701 Tf 10.583 -1.689 Td [(1)]TJ/F23 7.9701 Tf 4.234 0 Td [(L)]TJ/F15 10.9091 Tf 9.894 1.689 Td [(and)]TJ/F22 10.9091 Tf 21.212 0 Td [(M)]TJ/F23 7.9701 Tf 10.583 -1.689 Td [(C)]TJ/F15 10.9091 Tf 10.733 1.689 Td [(are)-333(symme)-1(tr)1(ic.)]TJ -75.151 -13.892 Td [(As)-533(p)1(reviously)-533(noted,)-582(on)-533(parallel)-532(computers)-533(the)-532(n)28(um)28(b)-28(er)-533(of)-532(submatrices)-533(usually)]TJ -16.937 -13.549 Td 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[(,)-320(Chapter)-317(3]\051.)-439(Additiv)28(e)-318(and)-317(h)28(ybrid)-317(m)28(ultilev)28(el)-317(preconditioners)-317(are)-317(obtained)-317(as)-318(direct)]TJ 0 -13.549 Td [(extensions)-359(of)-359(the)-359(t)28(w)27(o-lev)28(el)-359(coun)28(terparts.)-522(F)84(or)-359(a)-359(detailed)-359(descrition)-359(of)-359(them,)-366(the)-359(reader)]TJ 0 -13.55 Td [(is)-321(referred)-321(to)-321([)]TJ -1 0 0 rg 1 0 0 RG - [(23)]TJ -0 g 0 G - [(,)-324(Chap)1(te)-1(r)-321(3].)-440(The)-321(algorithm)-321(for)-321(the)-321(application)-321(of)-321(a)-321(m)28(ulti-lev)28(el)-321(h)28(ybrid)]TJ 0 -13.549 Td [(p)-28(ost-smo)-28(othed)-335(preconditioner)]TJ/F22 10.9091 Tf 145.676 0 Td [(M)]TJ/F15 10.9091 Tf 15.429 0 Td [(to)-335(a)-335(v)28(ec)-1(tor)]TJ/F22 10.9091 Tf 55.241 0 Td [(v)]TJ/F15 10.9091 Tf 5.679 0 Td [(,)-336(i.e.)-335(for)-335(the)-335(computation)-335(of)]TJ/F22 10.9091 Tf 135.911 0 Td [(w)]TJ/F15 10.9091 Tf 11.166 0 Td [(=)]TJ/F22 10.9091 Tf 11.548 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 3.959 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 4.733 -3.959 Td [(v)]TJ/F15 10.9091 Tf 5.679 0 Td [(,)]TJ -409.421 -13.549 Td [(is)-425(rep)-28(orted,)-448(for)-425(example,)-448(in)-425(Figure)]TJ -0 0 1 rg 0 0 1 RG - [-425(1)]TJ -0 g 0 G - [(.)-720(Here)-425(the)-425(n)28(um)27(b)-27(er)-425(of)-426(lev)28(els)-425(is)-425(denoted)-425(b)28(y)]TJ/F22 10.9091 Tf 391.675 0 Td [(nl)-20(ev)]TJ/F15 10.9091 Tf -391.675 -13.549 Td [(and)-410(the)-411(lev)28(els)-411(are)-410(n)28(um)27(b)-27(ered)-411(in)-410(increasing)-411(order)-410(starting)-411(from)-410(the)-411(\014nest)-410(one,)-430(i.e.)-410(the)]TJ 0 -13.549 Td [(\014nest)-361(lev)28(el)-361(is)-362(lev)28(el)-361(1;)-375(the)-361(coarse)-361(matrix)-361(and)-361(the)-361(corresp)-28(onding)-361(basic)-361(preconditioner)-361(at)]TJ 0 -13.55 Td [(eac)28(h)-415(lev)28(el)]TJ/F22 10.9091 Tf 51.17 0 Td [(l)]TJ/F15 10.9091 Tf 7.995 0 Td [(are)-415(denoted)-414(b)27(y)]TJ/F22 10.9091 Tf 77.24 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 -1.777 Td [(l)]TJ/F15 10.9091 Tf 7.645 1.777 Td [(and)]TJ/F22 10.9091 Tf 22.1 0 Td [(M)]TJ/F23 7.9701 Tf 10.583 -1.777 Td [(l)]TJ/F15 10.9091 Tf 3.121 1.777 Td [(,)-435(resp)-28(ectiv)28(ely)83(,)-435(with)]TJ/F22 10.9091 Tf 96.23 0 Td [(A)]TJ/F20 7.9701 Tf 8.182 -1.636 Td [(1)]TJ/F15 10.9091 Tf 9.242 1.636 Td [(=)]TJ/F22 10.9091 Tf 12.996 0 Td [(A)]TJ/F15 10.9091 Tf 8.182 0 Td [(,)-435(while)-415(the)-415(r)1(e)-1(lated)]TJ -322.868 -13.549 Td [(restriction)-333(op)-28(erator)-333(is)-334(denoted)-333(b)28(y)]TJ/F22 10.9091 Tf 163.576 0 Td [(R)]TJ/F23 7.9701 Tf 8.283 -1.777 Td [(l)]TJ/F15 10.9091 Tf 3.12 1.777 Td [(.)]TJ/F17 11.9552 Tf -174.979 -31.172 Td [(4.2)-1125(Smo)-31(othed)-375(Aggregation)]TJ/F15 10.9091 Tf 0 -21.261 Td [(In)-411(order)-411(to)-411(de\014ne)-411(the)-411(restriction)-411(op)-27(erator)]TJ/F22 10.9091 Tf 207.99 0 Td [(R)]TJ/F23 7.9701 Tf 8.284 -1.689 Td [(C)]TJ/F15 10.9091 Tf 7.096 1.689 Td [(,)-430(whic)27(h)-411(is)-411(used)-411(to)-411(compute)-411(th)1(e)-412(coarse-)]TJ -223.37 -13.549 Td [(lev)28(el)-281(matrix)]TJ/F22 10.9091 Tf 59.188 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 -1.689 Td [(C)]TJ/F15 10.9091 Tf 7.097 1.689 Td [(,)-291(MLD2P4)-281(uses)-281(the)]TJ/F18 10.9091 Tf 93.551 0 Td [(smo)51(othe)51(d)-309(aggr)51(e)51(gation)]TJ/F15 10.9091 Tf 103.876 0 Td [(algorithm)-281(describ)-27(e)-1(d)-280(in)-281([)]TJ -1 0 0 rg 1 0 0 RG - [(1)]TJ -0 g 0 G - [(,)]TJ -1 0 0 rg 1 0 0 RG - [-281(27)]TJ -0 g 0 G - [(].)]TJ -271.894 -13.549 Td [(The)-443(basic)-442(idea)-443(of)-443(this)-442(algorithm)-443(is)-442(to)-443(build)-443(a)-442(coarse)-443(set)-443(of)-442(v)27(ertices)]TJ/F22 10.9091 Tf 335.939 0 Td [(W)]TJ/F23 7.9701 Tf 10.303 -1.689 Td [(C)]TJ/F15 10.9091 Tf 11.926 1.689 Td [(b)28(y)-443(suitably)]TJ -358.168 -13.55 Td [(grouping)-329(the)-330(v)28(ertices)-330(of)]TJ/F22 10.9091 Tf 116.02 0 Td [(W)]TJ/F15 10.9091 Tf 15.414 0 Td [(in)28(to)-330(disjoin)28(t)-329(subse)-1(t)1(s)-330(\050aggregates\051,)-331(and)-329(to)-330(de\014ne)-329(the)-330(coarse-)]TJ -131.434 -13.549 Td [(to-\014ne)-253(s)-1(p)1(ac)-1(e)-253(transfer)-253(op)-28(erator)]TJ/F22 10.9091 Tf 144.031 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 3.959 Td [(T)]TJ -0.084 -7.192 Td [(C)]TJ/F15 10.9091 Tf 9.862 3.233 Td [(b)28(y)-254(apply)1(ing)-254(a)-253(suitable)-254(smo)-28(other)-253(to)-254(a)-253(simple)-254(piecewise)]TJ -162.177 -13.549 Td [(constan)28(t)-334(pr)1(olongation)-334(op)-27(erator,)-334(to)-333(impro)28(v)28(e)-334(the)-333(qualit)28(y)-333(of)-334(the)-333(coarse-space)-334(correction.)]TJ 16.937 -13.892 Td [(Three)-333(main)-334(steps)-333(can)-333(b)-28(e)-333(iden)27(ti\014ed)-333(in)-333(the)-334(smo)-27(othed)-334(aggregation)-333(pro)-28(cedure:)]TJ -0 g 0 G -0 g 0 G -ET - -endstream -endobj -349 0 obj -<< -/Length 10394 ->> -stream -0 g 0 G -BT -/F15 10.9091 Tf 86.4 740.002 Td [(14)]TJ/F41 10.9091 Tf 203.265 0 Td [(MLD2P4)-378(User)67('s)-378(and)-378(Ref)1(erence)-378(Guide)]TJ 0 g 0 G 0 g 0 G 0 g 0 G 0 g 0 G -0 g 0 G -0 g 0 G -/F22 9.9626 Tf -124.174 -45.783 Td [(v)]TJ/F7 6.9738 Tf 4.829 -1.495 Td [(1)]TJ/F15 9.9626 Tf 7.237 1.495 Td [(=)]TJ/F22 9.9626 Tf 10.516 0 Td [(v)]TJ/F15 9.9626 Tf 5.186 0 Td [(;)]TJ/F43 9.9626 Tf -27.768 -17.625 Td [(for)]TJ/F22 9.9626 Tf 17.766 0 Td [(l)]TJ/F15 9.9626 Tf 5.936 0 Td [(=)-278(2)]TJ/F22 9.9626 Tf 15.498 0 Td [(;)-167(nl)-19(ev)]TJ/F43 9.9626 Tf 27.221 0 Td [(do)]TJ/F15 9.9626 Tf -56.459 -14.789 Td [(!)-444(transfer)]TJ/F22 9.9626 Tf 44.113 0 Td [(v)]TJ/F10 6.9738 Tf 4.829 -1.495 Td [(l)]TJ/F13 6.9738 Tf 2.56 0 Td [(\000)]TJ/F7 6.9738 Tf 6.227 0 Td [(1)]TJ/F15 9.9626 Tf 7.79 1.495 Td [(to)-333(the)-334(next)-333(coarser)-333(lev)27(el)]TJ/F22 9.9626 Tf -65.519 -11.956 Td [(v)]TJ/F10 6.9738 Tf 4.83 -1.494 Td [(l)]TJ/F15 9.9626 Tf 5.825 1.494 Td [(=)]TJ/F22 9.9626 Tf 10.516 0 Td [(R)]TJ/F10 6.9738 Tf 7.565 -1.494 Td [(l)]TJ/F22 9.9626 Tf 3.058 1.494 Td [(v)]TJ/F10 6.9738 Tf 4.829 -1.494 Td [(l)]TJ/F13 6.9738 Tf 2.56 0 Td [(\000)]TJ/F7 6.9738 Tf 6.226 0 Td [(1)]TJ/F43 9.9626 Tf -55.371 -13.295 Td [(endfor)]TJ/F15 9.9626 Tf 0 -17.625 Td [(!)-444(apply)-334(the)-333(coarsest-lev)28(el)-334(correction)]TJ/F22 9.9626 Tf 0 -15.177 Td [(y)]TJ/F10 6.9738 Tf 4.884 -1.494 Td [(nl)-6(ev)]TJ/F15 9.9626 Tf 18.766 1.494 Td [(=)]TJ/F22 9.9626 Tf 10.516 0 Td [(A)]TJ/F13 6.9738 Tf 7.472 4.262 Td [(\000)]TJ/F7 6.9738 Tf 6.227 0 Td [(1)]TJ/F10 6.9738 Tf -6.227 -7.267 Td [(nl)-6(ev)]TJ/F22 9.9626 Tf 15.998 3.005 Td [(v)]TJ/F10 6.9738 Tf 4.829 -1.494 Td [(nl)-6(ev)]TJ/F43 9.9626 Tf -62.465 -16.131 Td [(for)]TJ/F22 9.9626 Tf 17.767 0 Td [(l)]TJ/F15 9.9626 Tf 5.936 0 Td [(=)]TJ/F22 9.9626 Tf 10.516 0 Td [(nl)-20(ev)]TJ/F25 9.9626 Tf 21.188 0 Td [(\000)]TJ/F15 9.9626 Tf 9.962 0 Td [(1)]TJ/F22 9.9626 Tf 4.982 0 Td [(;)]TJ/F15 9.9626 Tf 4.427 0 Td [(1)]TJ/F22 9.9626 Tf 4.982 0 Td [(;)]TJ/F25 9.9626 Tf 4.427 0 Td [(\000)]TJ/F15 9.9626 Tf 7.749 0 Td [(1)]TJ/F43 9.9626 Tf 8.801 0 Td [(do)]TJ/F15 9.9626 Tf -90.774 -14.789 Td [(!)-444(transfer)]TJ/F22 9.9626 Tf 44.112 0 Td [(y)]TJ/F10 6.9738 Tf 4.885 -1.495 Td [(l)]TJ/F7 6.9738 Tf 2.559 0 Td [(+1)]TJ/F15 9.9626 Tf 13.907 1.495 Td [(to)-333(the)-334(next)-333(\014ner)-333(lev)28(el)]TJ/F22 9.9626 Tf -65.463 -11.968 Td [(y)]TJ/F10 6.9738 Tf 4.884 -1.494 Td [(l)]TJ/F15 9.9626 Tf 5.825 1.494 Td [(=)]TJ/F22 9.9626 Tf 10.517 0 Td [(R)]TJ/F10 6.9738 Tf 7.641 3.616 Td [(T)]TJ -0.077 -6.437 Td [(l)]TJ/F7 6.9738 Tf 2.56 0 Td [(+1)]TJ/F22 9.9626 Tf 10.585 2.821 Td [(y)]TJ/F10 6.9738 Tf 4.885 -1.494 Td [(l)]TJ/F7 6.9738 Tf 2.56 0 Td [(+1)]TJ/F15 9.9626 Tf 10.585 1.494 Td [(;)]TJ -59.965 -14.855 Td [(!)-444(compute)-334(the)-333(residual)-333(at)-334(the)-333(curren)28(t)-334(lev)28(el)]TJ/F22 9.9626 Tf 0 -12.342 Td [(r)]TJ/F10 6.9738 Tf 4.494 -1.495 Td [(l)]TJ/F15 9.9626 Tf 5.826 1.495 Td [(=)]TJ/F22 9.9626 Tf 10.516 0 Td [(v)]TJ/F10 6.9738 Tf 4.829 -1.495 Td [(l)]TJ/F25 9.9626 Tf 5.272 1.495 Td [(\000)]TJ/F22 9.9626 Tf 9.962 0 Td [(A)]TJ/F13 6.9738 Tf 7.472 4.262 Td [(\000)]TJ/F7 6.9738 Tf 6.227 0 Td [(1)]TJ/F10 6.9738 Tf -6.227 -7.268 Td [(l)]TJ/F22 9.9626 Tf 10.696 3.006 Td [(y)]TJ/F10 6.9738 Tf 4.885 -1.495 Td [(l)]TJ/F15 9.9626 Tf 3.058 1.495 Td [(;)]TJ -67.01 -14.79 Td [(!)-444(apply)-334(the)-333(basic)-333(Sc)27(h)28(w)28(arz)-333(preconditioner)-334(to)-333(the)-333(residual)]TJ/F22 9.9626 Tf 0 -12.343 Td [(r)]TJ/F10 6.9738 Tf 4.494 -1.494 Td [(l)]TJ/F15 9.9626 Tf 5.826 1.494 Td [(=)]TJ/F22 9.9626 Tf 10.516 0 Td [(M)]TJ/F13 6.9738 Tf 10.751 4.262 Td [(\000)]TJ/F7 6.9738 Tf 6.227 0 Td [(1)]TJ/F10 6.9738 Tf -7.313 -7.267 Td [(l)]TJ/F22 9.9626 Tf 11.782 3.005 Td [(r)]TJ/F10 6.9738 Tf 4.495 -1.494 Td [(l)]TJ/F15 9.9626 Tf -46.778 -13.295 Td [(!)-444(up)-28(date)]TJ/F22 9.9626 Tf 40.681 0 Td [(y)]TJ/F10 6.9738 Tf 4.884 -1.495 Td [(l)]TJ/F22 9.9626 Tf -45.565 -10.461 Td [(y)]TJ/F10 6.9738 Tf 4.884 -1.494 Td [(l)]TJ/F15 9.9626 Tf 5.826 1.494 Td [(=)]TJ/F22 9.9626 Tf 10.516 0 Td [(y)]TJ/F10 6.9738 Tf 4.884 -1.494 Td [(l)]TJ/F15 9.9626 Tf 5.272 1.494 Td [(+)]TJ/F22 9.9626 Tf 9.963 0 Td [(r)]TJ/F10 6.9738 Tf 4.494 -1.494 Td [(l)]TJ/F43 9.9626 Tf -55.802 -10.461 Td [(endfor)]TJ/F22 9.9626 Tf 0 -14.79 Td [(w)]TJ/F15 9.9626 Tf 10.168 0 Td [(=)]TJ/F22 9.9626 Tf 10.516 0 Td [(y)]TJ/F7 6.9738 Tf 4.884 -1.494 Td [(1)]TJ/F15 9.9626 Tf 4.47 1.494 Td [(;)]TJ + -292.247 -42.833 Td [(pro)-28(cedure)-333(V-cycle)]TJ/F28 10.9091 Tf 85.818 8.837 Td [(\000)]TJ/F22 10.9091 Tf 5 -8.837 Td [(k)-32(;)-166(A)]TJ/F23 7.9701 Tf 19.053 3.959 Td [(k)]TJ/F22 10.9091 Tf 5.12 -3.959 Td [(;)-167(b)]TJ/F23 7.9701 Tf 9.53 3.959 Td [(k)]TJ/F22 10.9091 Tf 5.12 -3.959 Td [(;)-167(u)]TJ/F23 7.9701 Tf 11.093 3.959 Td [(k)]TJ/F28 10.9091 Tf 5.12 4.878 Td [(\001)]TJ/F15 10.9091 Tf -134.945 -28.055 Td [(if)-333(\050)]TJ/F22 10.9091 Tf 14.242 0 Td [(k)]TJ/F25 10.9091 Tf 9.053 0 Td [(6)]TJ/F15 10.9091 Tf 0 0 Td [(=)]TJ/F22 10.9091 Tf 11.516 0 Td [(nl)-20(ev)]TJ/F15 10.9091 Tf 20.776 0 Td [(\051)-333(then)]TJ/F22 10.9091 Tf -44.678 -16.393 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.958 Td [(k)]TJ/F15 10.9091 Tf 8.15 -3.958 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.958 Td [(k)]TJ/F15 10.9091 Tf 7.544 -3.958 Td [(+)]TJ/F22 10.9091 Tf 10.909 0 Td [(M)]TJ/F23 7.9701 Tf 11.773 3.958 Td [(k)]TJ/F28 10.9091 Tf 6.938 4.878 Td [(\000)]TJ/F22 10.9091 Tf 5 -8.836 Td [(b)]TJ/F23 7.9701 Tf 4.682 3.958 Td [(k)]TJ/F25 10.9091 Tf 7.544 -3.958 Td [(\000)]TJ/F22 10.9091 Tf 10.909 0 Td [(A)]TJ/F23 7.9701 Tf 8.181 3.958 Td [(k)]TJ/F22 10.9091 Tf 5.12 -3.958 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.958 Td [(k)]TJ/F28 10.9091 Tf 5.12 4.878 Td [(\001)]TJ/F22 10.9091 Tf -122.12 -25.229 Td [(b)]TJ/F23 7.9701 Tf 4.682 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F15 10.9091 Tf 14.349 -3.958 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F28 10.9091 Tf 13.137 4.878 Td [(\000)]TJ/F22 10.9091 Tf 5 -8.836 Td [(b)]TJ/F23 7.9701 Tf 4.682 3.958 Td [(k)]TJ/F25 10.9091 Tf 7.544 -3.958 Td [(\000)]TJ/F22 10.9091 Tf 10.909 0 Td [(A)]TJ/F23 7.9701 Tf 8.181 3.958 Td [(k)]TJ/F22 10.9091 Tf 5.12 -3.958 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.958 Td [(k)]TJ/F28 10.9091 Tf 5.12 4.878 Td [(\001)]TJ/F22 10.9091 Tf -114.095 -25.229 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F15 10.9091 Tf 14.349 -3.958 Td [(=)-333(V-cycle)]TJ/F28 10.9091 Tf 47.273 8.836 Td [(\000)]TJ/F22 10.9091 Tf 5 -8.836 Td [(k)]TJ/F15 10.9091 Tf 8.447 0 Td [(+)-222(1)]TJ/F22 10.9091 Tf 16.363 0 Td [(;)-167(A)]TJ/F23 7.9701 Tf 13.03 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F22 10.9091 Tf 11.319 -3.958 Td [(;)-167(b)]TJ/F23 7.9701 Tf 9.53 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F22 10.9091 Tf 11.319 -3.958 Td [(;)]TJ/F15 10.9091 Tf 4.848 0 Td [(0)]TJ/F28 10.9091 Tf 5.455 8.836 Td [(\001)]TJ/F22 10.9091 Tf -167.044 -25.229 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.958 Td [(k)]TJ/F15 10.9091 Tf 8.15 -3.958 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.958 Td [(k)]TJ/F15 10.9091 Tf 7.544 -3.958 Td [(+)]TJ/F22 10.9091 Tf 10.909 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F22 10.9091 Tf 11.319 -3.958 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.621 0 Td [(+1)]TJ/F22 10.9091 Tf -85.934 -20.352 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.15 -3.959 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.959 Td [(k)]TJ/F15 10.9091 Tf 7.544 -3.959 Td [(+)]TJ/F22 10.9091 Tf 10.909 0 Td [(M)]TJ/F23 7.9701 Tf 11.773 3.959 Td [(k)]TJ/F28 10.9091 Tf 6.938 4.878 Td [(\000)]TJ/F22 10.9091 Tf 5 -8.837 Td [(b)]TJ/F23 7.9701 Tf 4.682 3.959 Td [(k)]TJ/F25 10.9091 Tf 7.544 -3.959 Td [(\000)]TJ/F22 10.9091 Tf 10.909 0 Td [(A)]TJ/F23 7.9701 Tf 8.181 3.959 Td [(k)]TJ/F22 10.9091 Tf 5.12 -3.959 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.959 Td [(k)]TJ/F28 10.9091 Tf 5.12 4.878 Td [(\001)]TJ/F15 10.9091 Tf -133.029 -25.22 Td [(else)]TJ/F22 10.9091 Tf 10.909 -18.375 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.15 -3.959 Td [(=)]TJ/F28 10.9091 Tf 11.515 8.836 Td [(\000)]TJ/F22 10.9091 Tf 5 -8.836 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ/F28 10.9091 Tf 5.12 4.877 Td [(\001)]TJ/F26 7.9701 Tf 5 -2.497 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 6.551 -6.339 Td [(b)]TJ/F23 7.9701 Tf 4.682 3.959 Td [(k)]TJ/F15 10.9091 Tf -77.94 -20.343 Td [(endif)]TJ 0 -16.393 Td [(return)]TJ/F22 10.9091 Tf 33.394 0 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.959 Td [(k)]TJ/F15 10.9091 Tf -50.548 -20.343 Td [(end)]TJ ET q -1 0 0 1 158.467 714.896 cm -[]0 d 0 J 0.398 w 0 0 m 268.316 0 l S +1 0 0 1 195.871 714.896 cm +[]0 d 0 J 0.398 w 0 0 m 207.909 0 l S Q q -1 0 0 1 158.666 443.282 cm -[]0 d 0 J 0.398 w 0 0 m 0 271.614 l S +1 0 0 1 196.071 504.814 cm +[]0 d 0 J 0.398 w 0 0 m 0 210.082 l S Q q -1 0 0 1 426.584 443.282 cm -[]0 d 0 J 0.398 w 0 0 m 0 271.614 l S +1 0 0 1 403.581 504.814 cm +[]0 d 0 J 0.398 w 0 0 m 0 210.082 l S Q q -1 0 0 1 158.467 443.282 cm -[]0 d 0 J 0.398 w 0 0 m 268.316 0 l S +1 0 0 1 195.871 504.814 cm +[]0 d 0 J 0.398 w 0 0 m 207.909 0 l S Q 0 g 0 G BT -/F15 10.9091 Tf 106.974 424.548 Td [(Figure)-333(1:)-445(Application)-333(of)-333(the)-333(m)27(ulti-lev)28(el)-333(h)28(ybrid)-334(p)-27(ost-smo)-28(othed)-334(p)1(rec)-1(on)1(ditioner.)]TJ +/F15 10.9091 Tf 165.28 486.08 Td [(Figure)-333(1:)-445(Application)-333(phase)-333(of)-333(a)-334(V-cycle)-333(preconditioner.)]TJ 0 g 0 G 0 g 0 G + -71.68 -47.165 Td [(op)-28(erator)]TJ/F22 10.9091 Tf 42.856 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ/F15 10.9091 Tf 7.611 -3.959 Td [(b)28(y)-229(apply)1(ing)-229(a)-228(suitable)-229(smo)-28(ot)1(he)-1(r)-228(to)-228(a)-229(simple)-228(piecewise)-229(constan)28(t)-228(prolongation)]TJ -58.986 -13.549 Td [(op)-28(erator,)-333(with)-333(the)-334(aim)-333(of)-333(impro)28(ving)-334(the)-333(qualit)28(y)-333(of)-334(the)-333(coarse-space)-334(correction.)]TJ 16.937 -14.651 Td [(Three)-333(main)-334(steps)-333(can)-333(b)-28(e)-334(i)1(den)27(ti\014ed)-333(in)-333(the)-334(smo)-27(othed)-334(aggregation)-333(pro)-28(cedure:)]TJ 0 g 0 G - -7.241 -42.981 Td [(1.)]TJ + -3.603 -26.924 Td [(1.)]TJ 0 g 0 G - [-500(coarsening)-333(of)-334(the)-333(v)28(ertex)-333(s)-1(et)]TJ/F22 10.9091 Tf 148.667 0 Td [(W)]TJ/F15 10.9091 Tf 11.818 0 Td [(,)-333(to)-334(obtain)]TJ/F22 10.9091 Tf 53.94 0 Td [(W)]TJ/F23 7.9701 Tf 10.303 -1.689 Td [(C)]TJ/F15 10.9091 Tf 7.096 1.689 Td [(;)]TJ + [-500(aggregation)-333(of)-334(t)1(he)-334(indices)-333(set)-334(\012)]TJ/F23 7.9701 Tf 164.697 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.119 -3.959 Td [(,)-333(to)-334(obtain)-333(\012)]TJ/F23 7.9701 Tf 61.819 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.621 0 Td [(+1)]TJ/F15 10.9091 Tf 11.319 -3.959 Td [(;)]TJ 0 g 0 G - -231.824 -21.526 Td [(2.)]TJ + -247.575 -26.924 Td [(2.)]TJ 0 g 0 G - [-500(construction)-333(of)-334(th)1(e)-334(prolongator)]TJ/F22 10.9091 Tf 166.515 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 3.959 Td [(T)]TJ -0.084 -7.192 Td [(C)]TJ/F15 10.9091 Tf 7.096 3.233 Td [(;)]TJ + [-500(construction)-333(of)-333(the)-334(prolongator)]TJ/F22 10.9091 Tf 166.515 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(;)]TJ 0 g 0 G - -181.895 -21.526 Td [(3.)]TJ -0 g 0 G - [-500(application)-333(of)]TJ/F22 10.9091 Tf 82.728 0 Td [(R)]TJ/F23 7.9701 Tf 8.283 -1.689 Td [(C)]TJ/F15 10.9091 Tf 10.733 1.689 Td [(and)]TJ/F22 10.9091 Tf 21.212 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.959 Td [(T)]TJ -0.084 -7.192 Td [(C)]TJ/F15 10.9091 Tf 10.733 3.233 Td [(to)-333(build)]TJ/F22 10.9091 Tf 41.212 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 -1.689 Td [(C)]TJ/F15 10.9091 Tf 7.097 1.689 Td [(.)]TJ -194.86 -20.041 Td [(T)83(o)-426(p)-28(erform)-426(the)-426(coarsening)-427(step,)-449(w)28(e)-427(ha)28(v)28(e)-426(implemen)27(ted)-426(the)-426(aggregation)-426(algorithm)]TJ -16.936 -13.549 Td [(sk)28(etc)27(h)1(e)-1(d)-388(in)-388([)]TJ -1 0 0 rg 1 0 0 RG - [(4)]TJ + -180.154 -26.923 Td [(3.)]TJ 0 g 0 G - [(].)-610(According)-389(to)-388([)]TJ + [-500(application)-333(of)]TJ/F22 10.9091 Tf 82.727 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.756 -3.959 Td [(and)]TJ/F22 10.9091 Tf 21.212 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.15 -3.959 Td [(=)-278(\050)]TJ/F22 10.9091 Tf 15.757 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(\051)]TJ/F23 7.9701 Tf 4.242 3.959 Td [(T)]TJ/F15 10.9091 Tf 10.241 -3.959 Td [(to)-333(build)]TJ/F22 10.9091 Tf 41.212 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F15 10.9091 Tf 11.319 -3.959 Td [(.)]TJ -243.343 -26.924 Td [(In)-407(order)-407(to)-407(p)-28(erform)-407(the)-407(coarsening)-407(s)-1(tep,)-425(the)-407(smo)-28(othed)-407(aggregation)-407(algorithm)-407(de-)]TJ -16.937 -13.549 Td [(scrib)-28(ed)-242(in)-243([)]TJ 1 0 0 rg 1 0 0 RG - [(27)]TJ + [(29)]TJ 0 g 0 G - [(],)-402(a)-389(mo)-28(di\014cation)-388(of)-389(this)-388(algorithm)-388(has)-389(b)-28(een)-388(actually)]TJ 0 -13.55 Td [(considered,)-257(in)-237(whic)28(h)-238(eac)28(h)-237(aggregate)]TJ/F22 10.9091 Tf 169.04 0 Td [(N)]TJ/F23 7.9701 Tf 8.765 -1.636 Td [(r)]TJ/F15 10.9091 Tf 7.145 1.636 Td [(is)-238(made)-237(of)-237(v)27(ertices)-237(of)]TJ/F22 10.9091 Tf 99.165 0 Td [(W)]TJ/F15 10.9091 Tf 14.409 0 Td [(that)-237(are)]TJ/F18 10.9091 Tf 39.756 0 Td [(str)51(ongly)-269(c)51(ouple)51(d)]TJ/F15 10.9091 Tf -338.28 -13.549 Td [(to)-333(a)-334(certain)-333(ro)-28(ot)-333(v)28(ertex)]TJ/F22 10.9091 Tf 115.242 0 Td [(r)]TJ/F25 10.9091 Tf 8.255 0 Td [(2)]TJ/F22 10.9091 Tf 10.303 0 Td [(W)]TJ/F15 10.9091 Tf 11.818 0 Td [(,)-333(i.e.)]TJ/F22 10.9091 Tf -39.683 -23.653 Td [(N)]TJ/F23 7.9701 Tf 8.765 -1.636 Td [(r)]TJ/F15 10.9091 Tf 7.585 1.636 Td [(=)]TJ/F28 10.9091 Tf 11.515 12.11 Td [(n)]TJ/F22 10.9091 Tf 7.273 -12.11 Td [(s)]TJ/F25 10.9091 Tf 8.144 0 Td [(2)]TJ/F22 10.9091 Tf 10.303 0 Td [(W)]TJ/F15 10.9091 Tf 14.848 0 Td [(:)]TJ/F25 10.9091 Tf 6.061 0 Td [(j)]TJ/F22 10.9091 Tf 3.03 0 Td [(a)]TJ/F23 7.9701 Tf 5.766 -1.636 Td [(r)-30(s)]TJ/F25 10.9091 Tf 8.471 1.636 Td [(j)]TJ/F22 10.9091 Tf 6.061 0 Td [(>)-278(\022)]TJ/F28 10.9091 Tf 16.939 9.86 Td [(p)]TJ + [(])-242(is)-243(used.)-414(In)-242(this)-243(algorithm,)-260(eac)27(h)-242(index)-243(in)-242(\012)]TJ/F23 7.9701 Tf 262.99 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.621 0 Td [(+1)]TJ/F15 10.9091 Tf 13.964 -3.959 Td [(corresp)-28(onds)-242(to)-243(an)-242(aggregate)]TJ -281.575 -13.549 Td [(of)-301(\012)]TJ/F23 7.9701 Tf 19.952 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.119 -3.959 Td [(,)-308(consisting)-301(of)-301(a)-301(suitably)-301(c)28(hosen)-301(index)]TJ/F22 10.9091 Tf 182.094 0 Td [(j)]TJ/F15 10.9091 Tf 8.402 0 Td [(and)-301(of)-301(the)-301(indices)]TJ/F22 10.9091 Tf 86.836 0 Td [(i)]TJ/F15 10.9091 Tf 7.043 0 Td [(that)-301(are)-301(strongly)-301(cou-)]TJ -309.446 -13.549 Td [(pled)-333(to)]TJ/F22 10.9091 Tf 36.97 0 Td [(j)]TJ/F15 10.9091 Tf 5.117 0 Td [(,)-333(i.e.,)]TJ/F25 10.9091 Tf 121.576 -19.945 Td [(j)]TJ/F22 10.9091 Tf 3.03 0 Td [(a)]TJ/F23 7.9701 Tf 5.767 4.504 Td [(k)]TJ 0 -7.201 Td [(ij)]TJ/F25 10.9091 Tf 7.265 2.697 Td [(j)]TJ/F22 10.9091 Tf 6.061 0 Td [(>)-278(\022)]TJ/F28 10.9091 Tf 16.939 12.648 Td [(q)]TJ ET q -1 0 0 1 318.005 264.251 cm -[]0 d 0 J 0.436 w 0 0 m 34.535 0 l S +1 0 0 1 307.234 255.295 cm +[]0 d 0 J 0.436 w 0 0 m 32.124 0 l S Q BT -/F25 10.9091 Tf 318.005 254.173 Td [(j)]TJ/F22 10.9091 Tf 3.03 0 Td [(a)]TJ/F23 7.9701 Tf 5.767 -1.636 Td [(r)-29(r)]TJ/F22 10.9091 Tf 8.611 1.636 Td [(a)]TJ/F23 7.9701 Tf 5.766 -1.636 Td [(ss)]TJ/F25 10.9091 Tf 8.33 1.636 Td [(j)]TJ/F28 10.9091 Tf 3.031 12.109 Td [(o)]TJ/F25 10.9091 Tf 9.697 -12.109 Td [([)-222(f)]TJ/F22 10.9091 Tf 15.151 0 Td [(r)]TJ/F25 10.9091 Tf 5.225 0 Td [(g)]TJ/F22 10.9091 Tf 7.273 0 Td [(;)]TJ/F15 10.9091 Tf -303.486 -24.259 Td [(for)-341(a)-340(giv)27(en)]TJ/F22 10.9091 Tf 54.517 0 Td [(\022)]TJ/F25 10.9091 Tf 8.59 0 Td [(2)]TJ/F15 10.9091 Tf 10.438 0 Td [([0)]TJ/F22 10.9091 Tf 8.485 0 Td [(;)]TJ/F15 10.9091 Tf 4.849 0 Td [(1].)-467(Since)-341(th)1(is)-341(algorithm)-341(has)-341(a)-341(sequen)28(tial)-341(n)1(ature,)-343(a)]TJ/F18 10.9091 Tf 242.976 0 Td [(de)51(c)51(ouple)51(d)]TJ/F15 10.9091 Tf 49.172 0 Td [(v)28(ersion)]TJ -379.027 -13.549 Td [(of)-399(it)-399(has)-399(b)-27(een)-399(c)27(hosen,)-415(where)-399(eac)28(h)-399(pro)-28(cessor)]TJ/F22 10.9091 Tf 216.236 0 Td [(i)]TJ/F15 10.9091 Tf 8.11 0 Td [(indep)-28(enden)28(tly)-399(applies)-399(the)-399(algor)1(ithm)-399(to)]TJ -224.346 -13.549 Td [(the)-421(set)-421(of)-421(v)28(ertices)]TJ/F22 10.9091 Tf 91.547 0 Td [(W)]TJ/F20 7.9701 Tf 11.818 3.959 Td [(0)]TJ/F23 7.9701 Tf -1.515 -7.014 Td [(i)]TJ/F15 10.9091 Tf 10.839 3.055 Td [(assigned)-421(to)-421(it)-421(in)-420(the)-421(initial)-421(data)-421(distribution.)-707(This)-420(v)27(ersion)-420(is)]TJ -112.689 -13.549 Td [(em)28(barrassingly)-250(parallel,)-267(since)-250(it)-251(do)-27(es)-251(not)-250(require)-250(an)28(y)-250(data)-250(c)-1(omm)28(unication.)-416(On)-251(th)1(e)-251(other)]TJ 0 -13.549 Td [(hand,)-312(it)-306(ma)28(y)-306(pro)-28(duce)-306(non-uniform)-306(aggregate)-1(s)-306(near)-306(b)-28(oundary)-306(v)28(ertices,)-312(i.e.)-306(near)-307(v)28(ertices)]TJ 0 -13.55 Td [(adjacen)28(t)-403(to)-403(v)27(ertices)-403(in)-403(other)-403(pro)-28(cessors,)-420(and)-403(is)-404(strongl)1(y)-404(dep)-27(enden)27(t)-403(on)-403(the)-403(n)28(um)28(b)-28(er)-403(of)]TJ 0 -13.549 Td [(pro)-28(cessors)-314(and)-314(on)-314(the)-314(initial)-314(partitioning)-314(of)-314(the)-314(matrix)]TJ/F22 10.9091 Tf 263.565 0 Td [(A)]TJ/F15 10.9091 Tf 8.182 0 Td [(.)-438(Nev)28(ertheless,)-318(this)-314(algorithm)]TJ -271.747 -13.549 Td [(has)-260(b)-27(een)-260(c)28(hosen)-260(for)-260(t)1(he)-260(implemen)28(tation)-260(in)-260(M)1(LD2P4,)-275(since)-259(it)-260(has)-260(b)-27(e)-1(en)-259(sho)28(wn)-260(to)-260(pro)-27(duce)]TJ 0 -13.549 Td [(go)-28(o)-28(d)-333(results)-333(in)-333(practice)-334([)]TJ +/F25 10.9091 Tf 307.234 242.428 Td [(j)]TJ/F22 10.9091 Tf 3.03 0 Td [(a)]TJ/F23 7.9701 Tf 5.767 3.758 Td [(k)]TJ 0 -7.015 Td [(ii)]TJ/F22 10.9091 Tf 6.264 3.257 Td [(a)]TJ/F23 7.9701 Tf 5.767 3.758 Td [(k)]TJ 0 -7.015 Td [(j)-58(j)]TJ/F25 10.9091 Tf 8.266 3.257 Td [(j)]TJ/F22 10.9091 Tf 3.03 0 Td [(;)]TJ/F15 10.9091 Tf -245.758 -26.063 Td [(for)-331(a)-331(giv)28(en)]TJ/F22 10.9091 Tf 54.197 0 Td [(\022)]TJ/F25 10.9091 Tf 8.455 0 Td [(2)]TJ/F15 10.9091 Tf 10.303 0 Td [([0)]TJ/F22 10.9091 Tf 8.484 0 Td [(;)]TJ/F15 10.9091 Tf 4.849 0 Td [(1].)-444(Since)-331(this)-331(algorith)1(m)-332(h)1(as)-332(a)-331(sequen)28(tial)-331(nature,)-331(a)-331(decoupled)-331(v)28(ersion)]TJ -86.288 -13.549 Td [(of)-306(it)-305(is)-306(applied,)-311(where)-306(eac)28(h)-306(pro)-28(cessor)]TJ/F22 10.9091 Tf 177.316 0 Td [(i)]TJ/F15 10.9091 Tf 7.094 0 Td [(indep)-28(enden)28(tly)-305(e)-1(xecutes)-305(the)-306(algorithm)-306(on)-305(the)-306(set)]TJ -184.41 -13.549 Td [(of)-341(i)1(ndices)-341(assigned)-341(to)-340(it)-341(in)-340(the)-341(initi)1(al)-341(data)-340(distribution.)-466(This)-341(v)28(ersion)-341(i)1(s)-341(em)28(barrassingly)]TJ 0 -13.549 Td [(parallel,)-506(since)-472(it)-472(do)-28(es)-472(not)-472(requ)1(ire)-472(an)28(y)-472(data)-472(comm)28(unication.)-860(On)-472(the)-472(other)-472(hand,)-506(it)]TJ 0 -13.55 Td [(ma)28(y)-394(pro)-27(duce)-394(some)-393(non-uniform)-393(aggre)-1(gat)1(e)-1(s)-393(and)-393(is)-394(strongly)-393(dep)-28(enden)28(t)-393(on)-394(the)-393(n)28(um)28(b)-28(er)]TJ 0 -13.549 Td [(of)-358(p)1(ro)-28(cessors)-358(and)-357(on)-358(the)-357(initial)-358(partitioning)-357(of)-358(the)-357(matrix)]TJ/F22 10.9091 Tf 280.52 0 Td [(A)]TJ/F15 10.9091 Tf 8.181 0 Td [(.)-517(Nev)28(ertheless,)-364(this)-357(parall)]TJ -288.701 -13.549 Td [(algorithm)-248(has)-249(b)-27(een)-249(c)28(hosen)-248(for)-249(MLD2P4,)-265(since)-248(it)-249(has)-248(b)-28(een)-248(sho)28(wn)-249(to)-248(pro)-28(duce)-248(go)-28(o)-28(d)-248(results)]TJ 0 -13.549 Td [(in)-333(practice)-334([)]TJ 1 0 0 rg 1 0 0 RG - [(3)]TJ + [(4)]TJ 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-333(4)]TJ + [-333(5)]TJ 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-333(26)]TJ + [-333(28)]TJ 0 g 0 G [(].)]TJ 0 g 0 G 0 g 0 G ET +endstream +endobj +356 0 obj +<< +/Length 14031 +>> +stream +0 g 0 G +BT +/F15 10.9091 Tf 86.4 740.002 Td [(14)]TJ/F41 10.9091 Tf 203.265 0 Td [(MLD2P4)-378(User)67('s)-378(and)-378(Ref)1(erence)-378(Guide)]TJ +0 g 0 G +/F15 10.9091 Tf -186.329 -35.866 Td [(The)-375(prolongator)]TJ/F22 10.9091 Tf 82.192 0 Td [(P)]TJ/F23 7.9701 Tf 8.518 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.216 -3.959 Td [(is)-375(built)-376(starting)-375(from)-375(a)-376(ten)28(tativ)28(e)-376(prolongator)]TJ 222.868 2.758 Td [(\026)]TJ/F22 10.9091 Tf -2.441 -2.758 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ/F25 10.9091 Tf 8.915 -3.959 Td [(2)]TJ/F34 10.9091 Tf 11.069 0 Td [(R)]TJ/F23 7.9701 Tf 7.878 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.139 -1.406 Td [(k)]TJ/F26 7.9701 Tf 4.573 1.406 Td [(\002)]TJ/F23 7.9701 Tf 6.586 0 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.406 Td [(k)]TJ/F21 5.9776 Tf 4.075 0 Td [(+1)]TJ/F15 10.9091 Tf 10.24 -2.553 Td [(,)]TJ -409.421 -13.549 Td [(de\014ned)-333(as)]TJ 115.609 -14.79 Td [(\026)]TJ/F22 10.9091 Tf -2.442 -2.757 Td [(P)]TJ/F23 7.9701 Tf 8.519 4.504 Td [(k)]TJ/F15 10.9091 Tf 8.15 -4.504 Td [(=)-278(\050)-85(\026)]TJ/F22 10.9091 Tf 15.758 0 Td [(p)]TJ/F23 7.9701 Tf 5.489 4.504 Td [(k)]TJ 0 -7.201 Td [(ij)]TJ/F15 10.9091 Tf 7.265 2.697 Td [(\051)]TJ/F22 10.9091 Tf 4.242 0 Td [(;)]TJ/F15 10.9091 Tf 16.684 0 Td [(\026)]TJ/F22 10.9091 Tf -0.926 0 Td [(p)]TJ/F23 7.9701 Tf 5.489 4.504 Td [(k)]TJ 0 -7.201 Td [(ij)]TJ/F15 10.9091 Tf 10.295 2.697 Td [(=)]TJ/F28 10.9091 Tf 11.515 15.382 Td [(\032)]TJ/F15 10.9091 Tf 13.163 -8.325 Td [(1)-1913(if)]TJ/F22 10.9091 Tf 35.721 0 Td [(i)]TJ/F25 10.9091 Tf 6.788 0 Td [(2)]TJ/F15 10.9091 Tf 10.303 0 Td [(\012)]TJ/F23 7.9701 Tf 7.879 3.959 Td [(k)]TJ 0 -7.014 Td [(j)]TJ/F22 10.9091 Tf 5.12 3.055 Td [(;)]TJ/F15 10.9091 Tf -65.811 -14.089 Td [(0)-1913(otherwise)]TJ/F22 10.9091 Tf 71.266 0 Td [(;)]TJ/F15 10.9091 Tf -290.076 -20.871 Td [(where)-357(\012)]TJ/F23 7.9701 Tf 39.687 3.958 Td [(k)]TJ 0 -7.014 Td [(j)]TJ/F15 10.9091 Tf 9.02 3.056 Td [(is)-357(the)-358(aggregate)-357(of)-358(\012)]TJ/F23 7.9701 Tf 100.235 3.958 Td [(k)]TJ/F15 10.9091 Tf 9.019 -3.958 Td [(corresp)-28(onding)-357(to)-358(the)-357(index)]TJ/F22 10.9091 Tf 132.69 0 Td [(j)]TJ/F25 10.9091 Tf 8.586 0 Td [(2)]TJ/F15 10.9091 Tf 10.742 0 Td [(\012)]TJ/F23 7.9701 Tf 7.879 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F15 10.9091 Tf 11.318 -3.958 Td [(.)]TJ/F22 10.9091 Tf 8.669 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.958 Td [(k)]TJ/F15 10.9091 Tf 9.02 -3.958 Td [(is)-357(obtained)]TJ -360.006 -15.096 Td [(b)28(y)-333(applying)-334(to)]TJ 75.471 2.758 Td [(\026)]TJ/F22 10.9091 Tf -2.441 -2.758 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.756 -3.959 Td [(a)-333(smo)-28(other)]TJ/F22 10.9091 Tf 56.758 0 Td [(S)]TJ/F23 7.9701 Tf 7.318 3.959 Td [(k)]TJ/F25 10.9091 Tf 8.15 -3.959 Td [(2)]TJ/F34 10.9091 Tf 10.303 0 Td [(R)]TJ/F23 7.9701 Tf 7.879 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.406 Td [(k)]TJ/F26 7.9701 Tf 4.573 1.406 Td [(\002)]TJ/F23 7.9701 Tf 6.587 0 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.406 Td [(k)]TJ/F15 10.9091 Tf 5.071 -2.553 Td [(:)]TJ/F22 10.9091 Tf -29.64 -24.189 Td [(P)]TJ/F23 7.9701 Tf 8.519 4.504 Td [(k)]TJ/F15 10.9091 Tf 8.15 -4.504 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(S)]TJ/F23 7.9701 Tf 7.318 4.504 Td [(k)]TJ/F15 10.9091 Tf 7.561 -1.746 Td [(\026)]TJ/F22 10.9091 Tf -2.441 -2.758 Td [(P)]TJ/F23 7.9701 Tf 8.519 4.504 Td [(k)]TJ/F22 10.9091 Tf 5.12 -4.504 Td [(;)]TJ/F15 10.9091 Tf -231.841 -24.189 Td [(in)-255(order)-255(to)-255(remo)27(v)28(e)-255(nonsmo)-28(oth)-255(comp)-28(onen)28(ts)-255(from)-255(the)-256(r)1(ange)-256(of)-255(the)-255(prolongator,)-271(and)-255(hence)]TJ 0 -13.549 Td [(to)-266(impro)27(v)28(e)-266(the)-267(con)28(v)28(ergence)-267(pr)1(op)-28(erties)-267(of)-266(the)-267(m)28(ulti-lev)28(el)-266(metho)-28(d)-267([)]TJ +1 0 0 rg 1 0 0 RG + [(1)]TJ +0 g 0 G + [(,)]TJ +1 0 0 rg 1 0 0 RG + [-266(27)]TJ +0 g 0 G + [(].)-422(A)-266(simple)-267(c)28(hoice)]TJ 0 -13.55 Td [(for)]TJ/F22 10.9091 Tf 16.697 0 Td [(S)]TJ/F23 7.9701 Tf 7.318 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.756 -3.959 Td [(is)-333(the)-334(damp)-27(ed)-334(Jacobi)-333(smo)-28(other:)]TJ/F22 10.9091 Tf 119.024 -24.189 Td [(S)]TJ/F23 7.9701 Tf 7.319 4.504 Td [(k)]TJ/F15 10.9091 Tf 8.15 -4.504 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(I)]TJ/F25 10.9091 Tf 8.076 0 Td [(\000)]TJ/F22 10.9091 Tf 10.909 0 Td [(!)]TJ/F23 7.9701 Tf 7.181 4.504 Td [(k)]TJ/F15 10.9091 Tf 5.12 -4.504 Td [(\050)]TJ/F22 10.9091 Tf 4.243 0 Td [(D)]TJ/F23 7.9701 Tf 9.334 4.504 Td [(k)]TJ/F15 10.9091 Tf 5.12 -4.504 Td [(\051)]TJ/F26 7.9701 Tf 4.243 4.504 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 4.732 -4.504 Td [(A)]TJ/F23 7.9701 Tf 8.182 4.504 Td [(k)]TJ/F22 10.9091 Tf 5.12 -4.504 Td [(;)]TJ/F15 10.9091 Tf -257.625 -24.189 Td [(where)]TJ/F22 10.9091 Tf 32.246 0 Td [(D)]TJ/F23 7.9701 Tf 9.335 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.457 -3.959 Td [(is)-398(the)-397(diagonal)-398(matrix)-397(with)-398(the)-397(sam)-1(e)-397(diagonal)-398(en)28(tries)-397(as)]TJ/F22 10.9091 Tf 278.83 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td 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Tf 2.884 0 Td [(=1)]TJ/F22 10.9091 Tf 13.665 13.087 Td [(P)]TJ/F23 7.9701 Tf 8.519 4.505 Td [(k)]TJ -1.515 -7.201 Td [(i)]TJ/F15 10.9091 Tf 6.635 2.696 Td [(\050)]TJ/F22 10.9091 Tf 4.242 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 4.505 Td [(k)]TJ 0 -7.201 Td [(i)]TJ/F15 10.9091 Tf 5.12 2.696 Td [(\051)]TJ/F26 7.9701 Tf 4.242 4.505 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F22 10.9091 Tf 4.732 -4.505 Td [(R)]TJ/F23 7.9701 Tf 8.368 4.505 Td [(k)]TJ -0.085 -7.201 Td [(i)]TJ/F22 10.9091 Tf 5.204 2.696 Td [(;)]TJ/F15 10.9091 Tf -272.298 -35.308 Td [(where)]TJ/F22 10.9091 Tf 31.036 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf 8.247 3.055 Td [(is)-287(supp)-27(osed)-287(to)-287(b)-27(e)-287(nonsingular.)-429(W)84(e)-287(observ)28(e)-287(that)-287(an)-286(appro)28(ximate)-287(in)28(v)28(e)-1(r)1(s)-1(e)-286(of)]TJ/F22 10.9091 Tf 351.684 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf -407.331 -10.494 Td [(is)-366(usually)-365(c)-1(on)1(s)-1(id)1(e)-1(r)1(e)-1(d)-365(instead)-366(of)-366(\050)]TJ/F22 10.9091 Tf 157.8 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf 5.12 3.055 Td [(\051)]TJ/F26 7.9701 Tf 4.242 3.959 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F15 10.9091 Tf 4.732 -3.959 Td [(.)-542(The)-366(setup)-365(of)]TJ/F22 10.9091 Tf 74.003 0 Td [(S)]TJ/F23 7.9701 Tf 7.318 3.959 Td [(k)]TJ -0.628 -7.192 Td [(AS)]TJ/F15 10.9091 Tf 16.427 3.233 Td [(during)-366(the)-365(m)27(ultil)1(e)-1(v)28(el)-366(bu)1(ild)]TJ -283.783 -13.549 Td [(phase)-333(in)28(v)27(olv)28(es)]TJ +0 g 0 G +/F25 10.9091 Tf 16.364 -22.25 Td [(\017)]TJ +0 g 0 G +/F15 10.9091 Tf 10.909 0 Td [(the)-383(de\014n)1(ition)-383(of)-383(th)1(e)-383(index)-383(subspaces)-382(\012)]TJ/F23 7.9701 Tf 188.861 3.959 Td [(k)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf 9.294 3.055 Td [(and)-382(of)-383(the)-383(corresp)-27(onding)-383(op)-28(erators)]TJ/F22 10.9091 Tf 173.536 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.959 Td [(k)]TJ -0.084 -7.014 Td [(i)]TJ/F15 10.9091 Tf -379.974 -10.494 Td [(\050and)]TJ/F22 10.9091 Tf 25.454 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ -1.515 -7.015 Td [(i)]TJ/F15 10.9091 Tf 6.635 3.056 Td [(\051;)]TJ +0 g 0 G +0 g 0 G +ET + endstream endobj 363 0 obj << -/Length 3919 +/Length 5723 >> stream 0 g 0 G @@ -2056,42 +2055,45 @@ stream BT /F41 10.9091 Tf 93.6 740.002 Td [(4)]TJ 0 g 0 G - [-378(Mul)67(ti-level)-378(Domain)-378(Decompo)1(siti)-1(o)1(n)-378(Ba)22(ck)22(gr)22(ound)]TJ/F15 10.9091 Tf 401.542 0 Td [(15)]TJ + [-378(Mul)67(tigrid)-378(Ba)22(ck)22(gr)23(ound)]TJ/F15 10.9091 Tf 401.542 0 Td [(15)]TJ 0 g 0 G - -384.605 -35.866 Td [(The)-277(prolongator)]TJ/F22 10.9091 Tf 80.038 0 Td [(P)]TJ/F23 7.9701 Tf 7.004 -1.688 Td [(C)]TJ/F15 10.9091 Tf 10.127 1.688 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.959 Td [(T)]TJ -0.084 -7.191 Td [(C)]TJ/F15 10.9091 Tf 10.116 3.232 Td [(is)-277(built)-276(starting)-277(from)-277(a)]TJ/F18 10.9091 Tf 109.52 0 Td [(tentative)-306(pr)51(olongator)]TJ/F22 10.9091 Tf 102.595 0 Td [(P)]TJ/F25 10.9091 Tf 11.549 0 Td [(2)-278(<)]TJ/F23 7.9701 Tf 18.182 3.959 Td [(n)]TJ/F26 7.9701 Tf 5.138 0 Td [(\002)]TJ/F23 7.9701 Tf 6.586 0 Td [(n)]TJ/F24 5.9776 Tf 5.139 -1.339 Td [(C)]TJ/F15 10.9091 Tf 6.692 -2.62 Td [(,)]TJ -409.421 -13.549 Td [(de\014ned)-333(as)]TJ/F22 10.9091 Tf 114.819 -17.818 Td [(P)]TJ/F15 10.9091 Tf 11.549 0 Td [(=)-278(\050)]TJ/F22 10.9091 Tf 15.757 0 Td [(p)]TJ/F23 7.9701 Tf 5.489 -1.637 Td [(ij)]TJ/F15 10.9091 Tf 7.265 1.637 Td [(\051)]TJ/F22 10.9091 Tf 4.243 0 Td [(;)-1167(p)]TJ/F23 7.9701 Tf 21.246 -1.637 Td [(ij)]TJ/F15 10.9091 Tf 10.296 1.637 Td [(=)]TJ/F28 10.9091 Tf 11.515 15.381 Td [(\032)]TJ/F15 10.9091 Tf 13.163 -9.135 Td [(1)-1913(if)]TJ/F22 10.9091 Tf 35.72 0 Td [(i)]TJ/F25 10.9091 Tf 6.789 0 Td [(2)]TJ/F22 10.9091 Tf 10.303 0 Td [(V)]TJ/F23 7.9701 Tf 8.787 5.307 Td [(j)]TJ -2.424 -8.741 Td [(C)]TJ/F15 10.9091 Tf -59.175 -10.116 Td [(0)-1913(otherwise)]TJ/F22 10.9091 Tf 79.261 7.304 Td [(:)]TJ 0 g 0 G -/F15 10.9091 Tf 103.909 0 Td [(\0502\051)]TJ +/F25 10.9091 Tf -385.178 -35.866 Td [(\017)]TJ 0 g 0 G -/F22 10.9091 Tf -398.512 -27.614 Td [(P)]TJ/F23 7.9701 Tf 7.004 -1.689 Td [(C)]TJ/F15 10.9091 Tf 10.733 1.689 Td [(is)-333(obtained)-334(b)28(y)-333(applying)-333(to)]TJ/F22 10.9091 Tf 128.849 0 Td [(P)]TJ/F15 10.9091 Tf 12.155 0 Td [(a)-333(smo)-28(other)]TJ/F22 10.9091 Tf 56.758 0 Td [(S)]TJ/F25 10.9091 Tf 10.348 0 Td [(2)-278(<)]TJ/F23 7.9701 Tf 18.182 3.958 Td [(n)]TJ/F26 7.9701 Tf 5.138 0 Td [(\002)]TJ/F23 7.9701 Tf 6.587 0 Td [(n)]TJ/F15 10.9091 Tf 5.636 -3.958 Td [(:)]TJ/F22 10.9091 Tf -79.224 -24.508 Td [(P)]TJ/F23 7.9701 Tf 7.004 -1.689 Td [(C)]TJ/F15 10.9091 Tf 10.127 1.689 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(S)-58(P)-27(;)]TJ +/F15 10.9091 Tf 10.909 0 Td [(the)-333(computation)-334(of)-333(the)-333(submatrices)]TJ/F22 10.9091 Tf 173.788 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf 5.12 3.055 Td [(;)]TJ 0 g 0 G -/F15 10.9091 Tf 187.7 0 Td [(\0503\051)]TJ +/F25 10.9091 Tf -197.999 -22.515 Td [(\017)]TJ 0 g 0 G - -398.512 -24.509 Td [(in)-303(order)-303(to)-303(remo)27(v)28(e)-303(oscillatory)-303(comp)-28(onen)28(ts)-303(from)-304(the)-303(range)-303(of)-303(the)-303(prolongator)-303(and)-303(hence)]TJ 0 -13.549 Td [(to)-433(impro)28(v)27(e)-433(the)-433(con)27(v)28(ergence)-433(prop)-28(erties)-433(of)-433(the)-434(m)28(ulti-lev)28(el)-434(S)1(c)27(h)28(w)28(arz)-433(m)-1(etho)-27(d)-434([)]TJ -1 0 0 rg 1 0 0 RG - [(1)]TJ +/F15 10.9091 Tf 10.909 0 Td [(the)-424(computation)-425(of)-424(their)-424(in)27(v)28(erses)-424(\050usually)-425(appro)28(ximated)-424(through)-424(some)-425(form)-424(of)]TJ 0 -13.549 Td [(incomplete)-333(factorization\051.)]TJ -27.273 -22.516 Td [(The)-410(computation)-411(of)]TJ/F22 10.9091 Tf 101.006 0 Td [(z)]TJ/F23 7.9701 Tf 5.553 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.551 -3.959 Td [(=)]TJ/F22 10.9091 Tf 12.916 0 Td [(M)]TJ/F23 7.9701 Tf 11.773 3.959 Td [(k)]TJ -1.19 -7.192 Td [(AS)]TJ/F22 10.9091 Tf 12.437 3.233 Td [(w)]TJ/F23 7.9701 Tf 8.104 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.119 -3.959 Td [(,)-430(with)]TJ/F22 10.9091 Tf 33.407 0 Td [(w)]TJ/F23 7.9701 Tf 8.103 3.959 Td [(k)]TJ/F25 10.9091 Tf 9.551 -3.959 Td [(2)]TJ/F34 10.9091 Tf 11.704 0 Td [(R)]TJ/F23 7.9701 Tf 7.878 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.139 -1.406 Td [(k)]TJ/F15 10.9091 Tf 5.071 -2.553 Td [(,)-430(dur)1(ing)-411(the)-410(m)28(ultilev)27(el)-410(application)]TJ -246.122 -13.549 Td [(phase,)-333(requires)]TJ 0 g 0 G - [(,)]TJ -1 0 0 rg 1 0 0 RG - [-433(25)]TJ +/F25 10.9091 Tf 16.364 -22.516 Td [(\017)]TJ +0 g 0 G +/F15 10.9091 Tf 10.909 0 Td [(the)-333(restriction)-334(of)]TJ/F22 10.9091 Tf 83.455 0 Td [(w)]TJ/F23 7.9701 Tf 8.103 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.756 -3.959 Td [(to)-333(the)-334(subspaces)]TJ/F34 10.9091 Tf 82 0 Td [(R)]TJ/F23 7.9701 Tf 7.879 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.406 Td [(k)-21(;i)]TJ/F15 10.9091 Tf 10.004 -2.553 Td [(,)-333(i.e.)]TJ/F22 10.9091 Tf 24.243 0 Td [(w)]TJ/F23 7.9701 Tf 8.103 3.959 Td [(k)]TJ -0.293 -7.014 Td [(i)]TJ/F15 10.9091 Tf 8.443 3.055 Td [(=)]TJ/F22 10.9091 Tf 11.516 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.959 Td [(k)]TJ -0.084 -7.014 Td [(i)]TJ/F22 10.9091 Tf 5.204 3.055 Td [(w)]TJ/F23 7.9701 Tf 8.103 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(;)]TJ 0 g 0 G - [(].)-744(A)]TJ 0 -13.549 Td [(simple)-333(c)27(hoice)-333(for)]TJ/F22 10.9091 Tf 83.122 0 Td [(S)]TJ/F15 10.9091 Tf 10.954 0 Td [(is)-333(the)-334(damp)-27(ed)-334(Jacobi)-333(smo)-28(other:)]TJ/F22 10.9091 Tf 71.293 -24.508 Td [(S)]TJ/F15 10.9091 Tf 10.348 0 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(I)]TJ/F25 10.9091 Tf 8.076 0 Td [(\000)]TJ/F22 10.9091 Tf 10.909 0 Td [(!)-36(D)]TJ/F26 7.9701 Tf 16.517 4.504 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 4.733 -4.504 Td [(A;)]TJ +/F25 10.9091 Tf -294.966 -22.515 Td [(\017)]TJ 0 g 0 G -/F15 10.9091 Tf 164.459 0 Td [(\0504\051)]TJ +/F15 10.9091 Tf 10.909 0 Td [(the)-333(computation)-334(of)-333(the)-333(v)28(ectors)]TJ/F22 10.9091 Tf 150.697 0 Td [(z)]TJ/F23 7.9701 Tf 5.553 3.958 Td [(k)]TJ -0.48 -7.014 Td [(i)]TJ/F15 10.9091 Tf 8.63 3.056 Td [(=)-278(\050)]TJ/F22 10.9091 Tf 15.758 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.958 Td [(k)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf 5.119 3.056 Td [(\051)]TJ/F26 7.9701 Tf 4.243 3.958 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 4.733 -3.958 Td [(w)]TJ/F23 7.9701 Tf 8.103 3.958 Td [(k)]TJ -0.293 -7.014 Td [(i)]TJ/F15 10.9091 Tf 5.413 3.056 Td [(;)]TJ 0 g 0 G - -398.512 -24.508 Td [(where)-305(the)-305(v)55(alue)-305(of)]TJ/F22 10.9091 Tf 89.707 0 Td [(!)]TJ/F15 10.9091 Tf 10.51 0 Td [(can)-305(b)-28(e)-305(c)28(hosen)-305(using)-305(some)-306(estimate)-305(of)-305(the)-305(sp)-28(ectral)-305(radius)-305(of)]TJ/F22 10.9091 Tf 283.399 0 Td [(D)]TJ/F26 7.9701 Tf 9.335 3.959 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 4.732 -3.959 Td [(A)]TJ/F15 10.9091 Tf -404.269 -13.549 Td [([)]TJ +/F25 10.9091 Tf -233.153 -22.516 Td [(\017)]TJ +0 g 0 G +/F15 10.9091 Tf 10.909 0 Td [(the)-333(prolongation)-333(and)-334(the)-333(sum)-333(of)-334(the)-333(previous)-333(v)28(e)-1(ctors,)-333(i.e.)]TJ/F22 10.9091 Tf 277.849 0 Td [(z)]TJ/F23 7.9701 Tf 5.553 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.15 -3.959 Td [(=)]TJ/F28 10.9091 Tf 11.515 8.182 Td [(P)]TJ/F23 7.9701 Tf 11.515 -3.019 Td [(m)]TJ/F24 5.9776 Tf 7.491 -1.405 Td [(k)]TJ/F23 7.9701 Tf -7.491 -7.015 Td [(i)]TJ/F20 7.9701 Tf 2.883 0 Td [(=1)]TJ/F22 10.9091 Tf 13.137 3.257 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ -1.515 -7.014 Td [(i)]TJ/F22 10.9091 Tf 6.635 3.055 Td [(z)]TJ/F23 7.9701 Tf 5.553 3.959 Td [(k)]TJ -0.48 -7.014 Td [(i)]TJ/F15 10.9091 Tf 5.6 3.055 Td [(.)]TJ -382.187 -22.515 Td [(V)83(arian)28(ts)-291(of)-291(the)-291(classical)-291(AS)-291(metho)-28(d,)-299(whic)28(h)-291(use)-291(mo)-28(di\014cations)-291(of)-291(the)-291(restriction)-291(and)-291(pro-)]TJ 0 -13.55 Td [(longation)-244(op)-28(erators,)-262(are)-244(also)-244(implemen)28(ted)-244(in)-244(MLD2P4.)-415(Among)-244(them,)-262(the)-244(Restricted)-244(AS)]TJ 0 -13.549 Td [(\050RAS\051)-375(preconditi)1(one)-1(r)-374(usually)-375(outp)-27(erforms)-375(the)-375(classical)-375(AS)-374(preconditioner)-375(in)-374(terms)-375(of)]TJ 0 -13.549 Td [(con)28(v)28(erge)-1(n)1(c)-1(e)-337(rate)-337(and)-338(of)-337(computation)-338(and)-337(comm)28(unication)-338(time)-337(on)-338(parallel)-337(distributed-)]TJ 0 -13.549 Td [(memory)-309(computers,)-314(and)-310(is)-309(therefore)-309(the)-310(most)-309(widely)-309(used)-309(am)-1(on)1(g)-310(the)-309(AS)-309(precondition-)]TJ 0 -13.549 Td [(ers)-333([)]TJ 1 0 0 rg 1 0 0 RG - [(1)]TJ + [(6)]TJ 0 g 0 G - [(].)]TJ + [(].)]TJ 16.937 -13.55 Td [(Direct)-418(solv)28(e)-1(rs)-418(based)-418(on)-418(sparse)-419(LU)-418(factorizations,)-439(implemen)27(ted)-418(in)-418(the)-418(third)-418(part)27(y)]TJ -16.937 -13.549 Td [(libraries)-374(rep)-27(orted)-374(in)-374(Section)]TJ +0 0 1 rg 0 0 1 RG + [-373(3.2)]TJ +0 g 0 G + [(,)-384(can)-374(b)-27(e)-374(applied)-374(as)-373(coarsest-lev)27(el)-373(solv)27(ers)-373(b)27(y)-373(MLD2P4.)]TJ 0 -13.549 Td [(Nativ)28(e)-313(inexact)-314(solv)28(ers)-313(based)-313(on)-313(incomplete)-314(LU)-313(factorizations,)-317(as)-313(w)28(ell)-314(as)-313(Jacobi,)-317(h)28(ybrid)]TJ 0 -13.549 Td [(\050forw)28(ard\051)-478(Gauss-Seidel,)-513(and)-478(blo)-28(c)28(k)-478(Jacobi)-477(preconditioners)-478(are)-477(also)-478(a)28(v)55(ailable.)-877(Direct)]TJ 0 -13.549 Td [(solv)28(ers)-279(usually)-278(lead)-279(to)-278(more)-279(e\013ectiv)28(e)-279(preconditi)1(oners)-279(in)-278(terms)-279(of)-278(algorithmic)-279(scalabilit)28(y;)]TJ 0 -13.55 Td [(ho)28(w)28(ev)27(er,)-333(this)-333(do)-28(es)-333(not)-334(guaran)28(tee)-333(parallel)-334(e\016ciency)84(.)]TJ 0 g 0 G 0 g 0 G ET endstream endobj -378 0 obj +376 0 obj << -/Length 8458 +/Length 8463 >> stream 0 g 0 G @@ -2100,7 +2102,7 @@ BT 0 g 0 G /F17 14.3462 Tf -203.265 -35.866 Td [(5)-1125(Getting)-375(Started)]TJ/F15 10.9091 Tf 0 -24.802 Td [(W)83(e)-441(describ)-28(e)-441(the)-442(basics)-441(for)-441(building)-442(an)1(d)-442(applying)-441(MLD2P4)-441(one-lev)27(el)-441(and)-441(m)27(u)1(lti-lev)27(el)]TJ 0 -13.549 Td [(\050i.e.,)-468(AMG\051)-441(precondition)1(e)-1(rs)-441(with)-440(the)-441(Krylo)27(v)-441(solv)28(ers)-441(included)-441(in)-441(P)1(SBLAS)-441([)]TJ 1 0 0 rg 1 0 0 RG - [(16)]TJ + [(17)]TJ 0 g 0 G [(].)-768(The)]TJ 0 -13.549 Td [(follo)28(wing)-333(steps)-334(are)-333(required:)]TJ 0 g 0 G @@ -2142,11 +2144,11 @@ BT 0 g 0 G 4.606 -23.442 Td [(4.1)]TJ 0 g 0 G -/F18 10.9091 Tf 19.394 0 Td [(Build)-328(the)-328(aggr)51(e)51(gation)-328(hier)51(ar)52(chy)-329(f)1(or)-329(a)-328(given)-328(matrix.)]TJ/F15 10.9091 Tf 244.224 0 Td [(This)-301(is)-301(p)-28(erformed)-301(b)28(y)-301(the)]TJ -244.224 -13.549 Td [(routine)]TJ/F44 10.9091 Tf 37.606 0 Td [(hierarchy_bld)]TJ/F15 10.9091 Tf 74.454 0 Td [(.)]TJ +/F18 10.9091 Tf 19.394 0 Td [(Build)-328(the)-328(aggr)51(e)51(gation)-328(hier)51(ar)52(chy)-329(f)1(or)-329(a)-328(given)-328(matrix.)]TJ/F15 10.9091 Tf 244.224 0 Td [(This)-301(is)-301(p)-28(erformed)-301(b)28(y)-301(the)]TJ -244.224 -13.549 Td [(routine)]TJ/F44 10.9091 Tf 37.606 0 Td [(hierarchy_build)]TJ/F15 10.9091 Tf 85.908 0 Td [(.)]TJ 0 g 0 G - -131.454 -17.997 Td [(4.2)]TJ + -142.908 -17.997 Td [(4.2)]TJ 0 g 0 G -/F18 10.9091 Tf 19.394 0 Td [(Build)-295(the)-295(pr)51(e)52(c)51(onditioner)-295(for)-295(a)-295(given)-295(matrix.)]TJ/F15 10.9091 Tf 208.934 0 Td [(This)-265(is)-265(p)-28(erformed)-265(b)28(y)-265(the)-265(routine)]TJ/F44 10.9091 Tf -208.934 -13.55 Td [(smoothers_bld)]TJ/F15 10.9091 Tf 74.453 0 Td [(.)]TJ -98.453 -23.442 Td [(If)-292(the)-292(selected)-292(preconditi)1(oner)-292(is)-292(one-lev)28(el,)-301(it)-291(is)-292(built)-292(in)-292(a)-292(single)-291(ste)-1(p)1(,)-301(p)-27(erformed)-292(b)28(y)]TJ 0 -13.549 Td [(the)-333(routine)]TJ/F44 10.9091 Tf 56.394 0 Td [(bld)]TJ/F15 10.9091 Tf 17.181 0 Td [(.)]TJ +/F18 10.9091 Tf 19.394 0 Td [(Build)-295(the)-295(pr)51(e)52(c)51(onditioner)-295(for)-295(a)-295(given)-295(matrix.)]TJ/F15 10.9091 Tf 208.934 0 Td [(This)-265(is)-265(p)-28(erformed)-265(b)28(y)-265(the)-265(routine)]TJ/F44 10.9091 Tf -208.934 -13.55 Td [(smoothers_build)]TJ/F15 10.9091 Tf 85.908 0 Td [(.)]TJ -109.908 -23.442 Td [(If)-292(the)-292(selected)-292(preconditi)1(oner)-292(is)-292(one-lev)28(el,)-301(it)-291(is)-292(built)-292(in)-292(a)-292(single)-291(ste)-1(p)1(,)-301(p)-27(erformed)-292(b)28(y)]TJ 0 -13.549 Td [(the)-333(routine)]TJ/F44 10.9091 Tf 56.394 0 Td [(bld)]TJ/F15 10.9091 Tf 17.181 0 Td [(.)]TJ 0 g 0 G -87.515 -23.442 Td [(5.)]TJ 0 g 0 G @@ -2161,9 +2163,9 @@ ET endstream endobj -397 0 obj +395 0 obj << -/Length 9294 +/Length 9271 >> stream 0 g 0 G @@ -2396,29 +2398,29 @@ BT 0 0 1 rg 0 0 1 RG [-339(1)]TJ 0 g 0 G - [(\051.)]TJ 0 -13.549 Td [(This)-318(preconditioner)-318(is)-318(c)28(hosen)-318(b)28(y)-318(s)-1(imp)1(ly)-318(s)-1(p)-27(ecifying)]TJ/F44 10.9091 Tf 242.529 0 Td [('ML')]TJ/F15 10.9091 Tf 26.378 0 Td [(as)-318(second)-318(argumen)28(t)-318(of)]TJ/F44 10.9091 Tf 109.181 0 Td [(P%init)]TJ/F15 10.9091 Tf -378.088 -13.549 Td [(\050a)-306(call)-305(to)]TJ/F44 10.9091 Tf 45.76 0 Td [(P%set)]TJ/F15 10.9091 Tf 31.97 0 Td [(is)-306(not)-305(needed\051)-306(and)-305(is)-306(applied)-306(with)-305(the)-306(CG)-305(solv)27(er)-305(pro)28(vided)-306(b)28(y)-306(PSBLAS)]TJ -77.73 -13.549 Td [(\050the)-271(m)-1(atr)1(ix)-272(of)-271(the)-272(system)-271(to)-272(b)-28(e)-271(solv)28(ed)-272(is)-271(ass)-1(u)1(m)-1(ed)-271(to)-271(b)-28(e)-272(p)-27(os)-1(i)1(tiv)27(e)-271(de\014nite\051.)-424(As)-271(previously)]TJ + [(\051.)]TJ 0 -13.549 Td [(This)-465(preconditioner)-464(is)-465(c)27(hosen)-464(b)27(y)-464(simply)-465(sp)-28(ecifying)]TJ/F44 10.9091 Tf 253.737 0 Td [('ML')]TJ/F15 10.9091 Tf 27.978 0 Td [(as)-465(the)-465(second)-464(argumen)27(t)-464(of)]TJ/F44 10.9091 Tf -281.715 -13.549 Td [(P%init)]TJ/F15 10.9091 Tf 38.192 0 Td [(\050a)-351(call)-351(to)]TJ/F44 10.9091 Tf 47.243 0 Td [(P%set)]TJ/F15 10.9091 Tf 32.464 0 Td [(is)-351(not)-351(needed\051)-351(and)-351(is)-351(appli)1(e)-1(d)-350(with)-351(the)-351(CG)-351(solv)28(er)-351(pro)27(vi)1(ded)-351(b)27(y)]TJ -117.899 -13.549 Td [(PSBLAS)-312(\050the)-311(m)-1(atr)1(ix)-312(of)-312(the)-312(system)-312(to)-311(b)-28(e)-312(solv)28(ed)-312(is)-312(assumed)-312(to)-312(b)-27(e)-312(p)-28(ositiv)28(e)-312(de\014nite\051.)-437(As)]TJ 0 g 0 G 0 g 0 G ET endstream endobj -411 0 obj +409 0 obj << -/Length 7797 +/Length 7782 >> stream 0 g 0 G BT /F15 10.9091 Tf 86.4 740.002 Td [(18)]TJ/F41 10.9091 Tf 203.265 0 Td [(MLD2P4)-378(User)67('s)-378(and)-378(Ref)1(erence)-378(Guide)]TJ 0 g 0 G -/F15 10.9091 Tf -203.265 -35.866 Td [(observ)28(ed,)-494(the)-461(mo)-28(dules)]TJ/F44 10.9091 Tf 114.087 0 Td [(psb_base_mod)]TJ/F15 10.9091 Tf 68.726 0 Td [(,)]TJ/F44 10.9091 Tf 8.414 0 Td [(mld_prec_mod)]TJ/F15 10.9091 Tf 73.76 0 Td [(and)]TJ/F44 10.9091 Tf 22.609 0 Td [(psb_krylov_mod)]TJ/F15 10.9091 Tf 85.215 0 Td [(m)28(ust)-462(b)-27(e)]TJ -372.811 -13.549 Td [(used)-333(b)28(y)-334(the)-333(example)-334(p)1(rogram.)]TJ 16.936 -13.549 Td [(The)-395(part)-395(of)-395(the)-395(co)-28(de)-395(concerning)-395(the)-395(reading)-395(and)-395(assem)27(blin)1(g)-396(of)-395(the)-395(sparse)-395(matrix)]TJ -16.936 -13.549 Td [(and)-457(the)-456(righ)27(t-han)1(d)-457(side)-457(v)28(ector,)-488(p)-28(erformed)-456(through)-457(the)-457(PSBLAS)-456(routines)-457(for)-457(sparse)]TJ 0 -13.549 Td [(matrix)-385(and)-385(v)28(ector)-386(managemen)28(t,)-398(is)-385(not)-385(rep)-28(orted)-385(here)-385(for)-385(brevit)28(y;)-412(t)1(he)-386(statemen)28(ts)-385(con-)]TJ 0 -13.55 Td [(cerning)-265(the)-264(deallo)-28(cation)-265(of)-264(the)-265(PSBLAS)-264(data)-265(structure)-265(are)-264(neglec)-1(ted)-264(to)-28(o.)-422(Th)1(e)-265(complete)]TJ 0 -13.549 Td [(co)-28(de)-306(can)-307(b)-27(e)-307(found)-306(in)-307(the)-306(example)-307(program)-306(\014le)]TJ/F44 10.9091 Tf 223.484 0 Td [(mld_dexample_ml.f90)]TJ/F15 10.9091 Tf 108.817 0 Td [(,)-312(in)-306(the)-307(directory)]TJ/F44 10.9091 Tf -332.301 -13.549 Td [(examples/fileread)]TJ/F15 10.9091 Tf 101.269 0 Td [(of)-358(the)-358(MLD2P4)-358(implemen)28(tation)-358(\050see)-359(Section)]TJ +/F15 10.9091 Tf -203.265 -35.866 Td [(previously)-352(observ)28(ed,)-357(the)-352(mo)-28(dules)]TJ/F44 10.9091 Tf 162.633 0 Td [(psb_base_mod)]TJ/F15 10.9091 Tf 68.726 0 Td [(,)]TJ/F44 10.9091 Tf 6.924 0 Td [(mld_prec_mod)]TJ/F15 10.9091 Tf 72.569 0 Td [(and)]TJ/F44 10.9091 Tf 21.418 0 Td [(psb_krylov_mod)]TJ/F15 10.9091 Tf -332.27 -13.549 Td [(m)28(ust)-334(b)-27(e)-334(used)-333(b)28(y)-334(t)1(he)-334(example)-333(program.)]TJ 16.936 -13.549 Td [(The)-395(part)-395(of)-395(the)-395(co)-28(de)-395(concerning)-395(the)-395(reading)-395(and)-395(assem)27(blin)1(g)-396(of)-395(the)-395(sparse)-395(matrix)]TJ -16.936 -13.549 Td [(and)-457(the)-456(righ)27(t-han)1(d)-457(side)-457(v)28(ector,)-488(p)-28(erformed)-456(through)-457(the)-457(PSBLAS)-456(routines)-457(for)-457(sparse)]TJ 0 -13.549 Td [(matrix)-385(and)-385(v)28(ector)-386(managemen)28(t,)-398(is)-385(not)-385(rep)-28(orted)-385(here)-385(for)-385(brevit)28(y;)-412(t)1(he)-386(statemen)28(ts)-385(con-)]TJ 0 -13.55 Td [(cerning)-265(the)-264(deallo)-28(cation)-265(of)-264(the)-265(PSBLAS)-264(data)-265(structure)-265(are)-264(neglec)-1(ted)-264(to)-28(o.)-422(Th)1(e)-265(complete)]TJ 0 -13.549 Td [(co)-28(de)-306(can)-307(b)-27(e)-307(found)-306(in)-307(the)-306(example)-307(program)-306(\014le)]TJ/F44 10.9091 Tf 223.484 0 Td [(mld_dexample_ml.f90)]TJ/F15 10.9091 Tf 108.817 0 Td [(,)-312(in)-306(the)-307(directory)]TJ/F44 10.9091 Tf -332.301 -13.549 Td [(examples/fileread)]TJ/F15 10.9091 Tf 101.269 0 Td [(of)-358(the)-358(MLD2P4)-358(implemen)28(tation)-358(\050see)-359(Section)]TJ 0 0 1 rg 0 0 1 RG [-358(3.5)]TJ 0 g 0 G [(\051.)-518(A)-358(sample)-358(tes)-1(t)]TJ -101.269 -13.549 Td [(problem)-375(along)-374(with)-375(the)-375(relev)56(an)28(t)-375(input)-374(data)-375(is)-375(a)28(v)56(ailable)-375(in)]TJ/F44 10.9091 Tf 283.422 0 Td [(examples/fileread/runs)]TJ/F15 10.9091 Tf 125.999 0 Td [(.)]TJ -409.421 -13.549 Td [(F)83(or)-333(details)-333(on)-334(th)1(e)-334(use)-333(of)-334(th)1(e)-334(PSBLAS)-333(routines,)-333(see)-334(the)-333(PSBLAS)-333(User's)-334(Guide)-333([)]TJ 1 0 0 rg 1 0 0 RG - [(16)]TJ + [(17)]TJ 0 g 0 G [(].)]TJ 16.936 -13.55 Td [(The)-497(setup)-496(and)-496(application)-497(of)-496(the)-496(default)-497(m)28(ulti-lev)28(el)-497(preconditioner)-496(for)-496(the)-497(real)]TJ -16.936 -13.549 Td [(single)-253(precision)-253(and)-253(the)-253(complex,)-269(single)-253(and)-253(double)-253(precision,)-269(v)27(ersions)-253(are)-253(obtained)-253(with)]TJ 0 -13.549 Td [(straigh)28(tforw)28(ard)-229(mo)-28(di\014cations)-229(of)-229(the)-228(previous)-229(example)-229(\050see)-229(Section)]TJ 0 0 1 rg 0 0 1 RG @@ -2428,11 +2430,11 @@ BT 0 0 1 rg 0 0 1 RG [-439(3)]TJ 0 g 0 G - 0 -13.549 Td [(sho)28(ws)-368(ho)28(w)-368(to)-368(set)-367(a)-368(V-cycle)-368(preconditioner)-367(whic)27(h)-367(applies)-368(1)-368(blo)-27(c)27(k-Jacobi)-367(sw)27(eep)-367(as)-368(pre-)]TJ 0 -13.549 Td [(and)-435(p)-27(ost-smo)-28(other,)-460(and)-435(solv)28(es)-435(the)-434(coarsest-le)-1(v)28(el)-434(system)-435(with)-435(8)-434(blo)-28(c)28(k-Jacobi)-435(sw)28(eeps.)]TJ 0 -13.549 Td [(Note)-442(that)-442(the)-442(ILU\0500\051)-442(factorization)-442(\050plus)-442(triangular)-442(solv)28(e\051)-442(is)-442(used)-443(as)-442(lo)-27(cal)-442(s)-1(olv)28(er)-442(for)]TJ 0 -13.549 Td [(the)-500(blo)-28(c)28(k-Jacobi)-501(sw)28(eeps,)-542(s)-1(i)1(nce)-501(this)-500(is)-501(the)-500(default)-501(asso)-28(ciated)-500(with)-500(blo)-28(c)28(k-Jacobi)-501(and)]TJ 0 -13.55 Td [(set)-449(b)28(y)]TJ/F44 10.9091 Tf 34.702 0 Td [(P%init)]TJ/F15 10.9091 Tf 34.364 0 Td [(.)-791(F)83(urth)1(e)-1(r)1(m)-1(or)1(e)-1(,)-477(sp)-28(ecifying)-449(blo)-28(c)28(k-Jacobi)-449(as)-449(coarsest-lev)28(el)-449(solv)28(er)-449(implies)]TJ -69.066 -13.549 Td [(that)-348(the)-347(c)-1(oarsest-lev)28(el)-348(matrix)-347(is)-348(distributed)-348(among)-348(the)-347(pro)-28(cesses.)-488(Figure)]TJ + 0 -13.549 Td [(sho)28(ws)-368(ho)28(w)-368(to)-368(set)-367(a)-368(V-cycle)-368(preconditioner)-367(whic)27(h)-367(applies)-368(1)-368(blo)-27(c)27(k-Jacobi)-367(sw)27(eep)-367(as)-368(pre-)]TJ 0 -13.549 Td [(and)-435(p)-27(ost-smo)-28(other,)-460(and)-435(solv)28(es)-435(the)-434(coarsest-le)-1(v)28(el)-434(system)-435(with)-435(8)-434(blo)-28(c)28(k-Jacobi)-435(sw)28(eeps.)]TJ 0 -13.549 Td [(Note)-442(that)-442(the)-442(ILU\0500\051)-442(factorization)-442(\050plus)-442(triangular)-442(solv)28(e\051)-442(is)-442(used)-443(as)-442(lo)-27(cal)-442(s)-1(olv)28(er)-442(for)]TJ 0 -13.549 Td [(the)-500(blo)-28(c)28(k-Jacobi)-501(sw)28(eeps,)-542(s)-1(i)1(nce)-501(this)-500(is)-501(the)-500(default)-501(asso)-28(ciated)-500(with)-500(blo)-28(c)28(k-Jacobi)-501(and)]TJ 0 -13.55 Td [(set)-449(b)28(y)]TJ/F44 10.9091 Tf 34.702 0 Td [(P%init)]TJ/F15 10.9091 Tf 34.364 0 Td [(.)-791(F)83(urth)1(e)-1(r)1(m)-1(ore,)-477(sp)-28(ecifying)-449(blo)-28(c)28(k-Jacobi)-449(as)-449(coarsest-lev)28(el)-449(solv)28(er)-449(implies)]TJ -69.066 -13.549 Td [(that)-348(the)-348(coarsest-lev)28(el)-348(matrix)-347(is)-348(distributed)-348(among)-348(the)-347(pro)-28(cesses.)-488(Figure)]TJ 0 0 1 rg 0 0 1 RG [-348(4)]TJ 0 g 0 G - [-348(sho)28(ws)-348(ho)28(w)]TJ 0 -13.549 Td [(to)-494(set)-493(a)-494(W-cycle)-493(preconditioner)-494(whic)28(h)-493(applies)-494(no)-493(pre-smo)-28(other)-494(and)-493(2)-494(Gauss-Seidel)]TJ 0 -13.549 Td [(sw)28(eeps)-303(as)-303(p)-28(ost-smo)-27(other,)-309(and)-303(solv)28(es)-303(the)-303(coarsest-lev)28(el)-303(system)-302(with)-303(the)-303(m)28(ultifron)28(tal)-303(LU)]TJ 0 -13.549 Td [(factorization)-377(implemen)28(ted)-377(in)-377(MUMPS.)-377(It)-377(is)-377(sp)-28(eci\014ed)-377(that)-377(the)-377(coarsest-lev)28(el)-377(matrix)-377(is)]TJ 0 -13.549 Td [(distributed,)-439(since)-417(MUMPS)-418(can)-418(b)-27(e)-418(used)-418(on)-418(b)-27(oth)-418(replicated)-418(and)-418(di)1(s)-1(tri)1(buted)-418(matrices,)]TJ 0 -13.55 Td [(and)-439(b)28(y)-438(default)-439(it)-439(is)-439(u)1(s)-1(ed)-438(on)-439(replicated)-439(ones.)-760(Note)-439(the)-439(use)-438(of)-439(the)-439(parameter)]TJ/F44 10.9091 Tf 380.786 0 Td [(pos)]TJ/F15 10.9091 Tf 21.968 0 Td [(to)]TJ -402.754 -13.549 Td [(sp)-28(ecify)-478(a)-477(prop)-28(ert)28(y)-478(only)-478(for)-477(the)-478(pre-smo)-28(other)-478(or)-477(the)-478(p)-28(ost-smo)-28(other)-477(\050see)-478(Section)]TJ + [-348(sho)28(ws)-348(ho)28(w)]TJ 0 -13.549 Td [(to)-494(set)-493(a)-494(W-cycle)-493(preconditioner)-494(whic)28(h)-493(applies)-494(no)-493(pre-smo)-28(other)-494(and)-493(2)-494(Gauss-Seidel)]TJ 0 -13.549 Td [(sw)28(eeps)-303(as)-303(p)-28(ost-smo)-27(other,)-309(and)-303(solv)28(es)-303(the)-303(coarsest-lev)28(el)-303(system)-302(with)-303(the)-303(m)28(ultifron)28(tal)-303(LU)]TJ 0 -13.549 Td [(factorization)-377(implemen)28(ted)-377(in)-377(MUMPS.)-377(It)-377(is)-377(sp)-28(eci\014ed)-377(that)-377(the)-377(coarsest-lev)28(el)-377(matrix)-377(is)]TJ 0 -13.549 Td [(distributed,)-439(since)-417(MUMPS)-418(can)-418(b)-27(e)-418(used)-418(on)-418(b)-27(oth)-418(replicated)-418(and)-418(distri)1(buted)-418(matrices,)]TJ 0 -13.55 Td [(and)-439(b)28(y)-438(default)-439(it)-439(is)-439(u)1(s)-1(ed)-438(on)-439(replicated)-439(ones.)-760(Note)-439(the)-439(use)-438(of)-439(the)-439(parameter)]TJ/F44 10.9091 Tf 380.786 0 Td [(pos)]TJ/F15 10.9091 Tf 21.968 0 Td [(to)]TJ -402.754 -13.549 Td [(sp)-28(ecify)-478(a)-477(prop)-28(ert)28(y)-478(only)-478(for)-477(the)-478(pre-smo)-28(other)-478(or)-477(the)-478(p)-28(ost-smo)-28(other)-477(\050see)-478(Section)]TJ 0 0 1 rg 0 0 1 RG [-478(6.2)]TJ 0 g 0 G @@ -2455,9 +2457,9 @@ ET endstream endobj -419 0 obj +417 0 obj << -/Length 3323 +/Length 3327 >> stream 0 g 0 G @@ -2472,7 +2474,7 @@ BT 0 g 0 G 0 g 0 G 0 g 0 G -/F44 9.9626 Tf -370.457 -30.995 Td [(use)-525(psb_base_mod)]TJ 0 -11.955 Td [(use)-525(mld_prec_mod)]TJ 0 -11.955 Td [(use)-525(psb_krylov_mod)]TJ -10.461 -11.956 Td [(...)-525(...)]TJ 0 -11.955 Td [(!)]TJ 0 -11.955 Td [(!)-525(sparse)-525(matrix)]TJ 10.461 -11.955 Td [(type\050psb_dspmat_type\051)-525(::)-525(A)]TJ -10.461 -11.955 Td [(!)-525(sparse)-525(matrix)-525(descriptor)]TJ 10.461 -11.955 Td [(type\050psb_desc_type\051)-1575(::)-525(desc_A)]TJ -10.461 -11.956 Td [(!)-525(preconditioner)]TJ 10.461 -11.955 Td [(type\050mld_dprec_type\051)-1050(::)-525(P)]TJ -10.461 -11.955 Td [(!)-525(right-hand)-525(side)-525(and)-525(solution)-525(vectors)]TJ 10.461 -11.955 Td [(type\050psb_d_vect_type\051)-525(::)-525(b,)-525(x)]TJ -10.461 -11.955 Td [(...)-525(...)]TJ 0 -11.955 Td [(!)]TJ 0 -11.956 Td [(!)-525(initialize)-525(the)-525(parallel)-525(environment)]TJ 10.461 -11.955 Td [(call)-525(psb_init\050ictxt\051)]TJ 0 -11.955 Td [(call)-525(psb_info\050ictxt,iam,np\051)]TJ -10.461 -11.955 Td [(...)-525(...)]TJ 0 -11.955 Td [(!)]TJ 0 -11.955 Td [(!)-525(read)-525(and)-525(assemble)-525(the)-525(spd)-525(matrix)-525(A)-525(and)-525(the)-525(right-hand)-525(side)-525(b)]TJ 0 -11.956 Td [(!)-525(using)-525(PSBLAS)-525(routines)-525(for)-525(sparse)-525(matrix)-525(/)-525(vector)-525(management)]TJ 0 -11.955 Td [(...)-525(...)]TJ 0 -11.955 Td [(!)]TJ 0 -11.955 Td [(!)-525(initialize)-525(the)-525(default)-525(multi-level)-525(preconditioner,)-525(i.e.)-525(V-cycle)]TJ 0 -11.955 Td [(!)-525(with)-525(basic)-525(smoothed)-525(aggregation,)-525(1)-525(hybrid)-525(forward/backward)]TJ 0 -11.956 Td [(!)-525(GS)-525(sweep)-525(as)-525(pre/post-smoother)-525(and)-525(UMFPACK)-525(as)-525(coarsest-level)]TJ 0 -11.955 Td [(!)-525(solver)]TJ 10.461 -11.955 Td [(call)-525(P%init\050P,'ML',info\051)]TJ -10.461 -11.955 Td [(!)]TJ 0 -11.955 Td [(!)-525(build)-525(the)-525(preconditioner)]TJ 10.461 -11.955 Td [(call)-525(P%hierarchy_bld\050A,desc_A,P,info\051)]TJ 0 -11.956 Td [(call)-525(P%smoothers_bld\050A,desc_A,P,info\051)]TJ -10.461 -23.91 Td [(!)]TJ 0 -11.955 Td [(!)-525(set)-525(the)-525(solver)-525(parameters)-525(and)-525(the)-525(initial)-525(guess)]TJ 10.461 -11.955 Td [(...)-525(...)]TJ -10.461 -11.955 Td [(!)]TJ 0 -11.956 Td [(!)-525(solve)-525(Ax=b)-525(with)-525(preconditioned)-525(CG)]TJ 10.461 -11.955 Td [(call)-525(psb_krylov\050'CG',A,P,b,x,tol,desc_A,info\051)]TJ 0 -11.955 Td [(...)-525(...)]TJ -10.461 -11.955 Td [(!)]TJ 0 -11.955 Td [(!)-525(deallocate)-525(the)-525(preconditioner)]TJ 10.461 -11.955 Td [(call)-525(P%free\050P,info\051)]TJ -10.461 -11.956 Td [(!)]TJ 0 -11.955 Td [(!)-525(deallocate)-525(other)-525(data)-525(structures)]TJ 10.461 -11.955 Td [(...)-525(...)]TJ -10.461 -11.955 Td [(!)]TJ 0 -11.955 Td [(!)-525(exit)-525(the)-525(parallel)-525(environment)]TJ 10.461 -11.956 Td [(call)-525(psb_exit\050ictxt\051)]TJ 0 -11.955 Td [(stop)]TJ/F15 10.9091 Tf -31.085 -21.354 Td [(Figure)-331(2:)-443(setup)-331(and)-331(application)-331(of)-331(the)-331(default)-331(m)28(ulti-lev)28(el)-331(preconditioner)-331(\050example)-331(1\051.)]TJ +/F44 9.9626 Tf -370.457 -30.995 Td [(use)-525(psb_base_mod)]TJ 0 -11.955 Td [(use)-525(mld_prec_mod)]TJ 0 -11.955 Td [(use)-525(psb_krylov_mod)]TJ -10.461 -11.956 Td [(...)-525(...)]TJ 0 -11.955 Td [(!)]TJ 0 -11.955 Td [(!)-525(sparse)-525(matrix)]TJ 10.461 -11.955 Td [(type\050psb_dspmat_type\051)-525(::)-525(A)]TJ -10.461 -11.955 Td [(!)-525(sparse)-525(matrix)-525(descriptor)]TJ 10.461 -11.955 Td [(type\050psb_desc_type\051)-1575(::)-525(desc_A)]TJ -10.461 -11.956 Td [(!)-525(preconditioner)]TJ 10.461 -11.955 Td [(type\050mld_dprec_type\051)-1050(::)-525(P)]TJ -10.461 -11.955 Td [(!)-525(right-hand)-525(side)-525(and)-525(solution)-525(vectors)]TJ 10.461 -11.955 Td [(type\050psb_d_vect_type\051)-525(::)-525(b,)-525(x)]TJ -10.461 -11.955 Td [(...)-525(...)]TJ 0 -11.955 Td [(!)]TJ 0 -11.956 Td [(!)-525(initialize)-525(the)-525(parallel)-525(environment)]TJ 10.461 -11.955 Td [(call)-525(psb_init\050ictxt\051)]TJ 0 -11.955 Td [(call)-525(psb_info\050ictxt,iam,np\051)]TJ -10.461 -11.955 Td [(...)-525(...)]TJ 0 -11.955 Td [(!)]TJ 0 -11.955 Td [(!)-525(read)-525(and)-525(assemble)-525(the)-525(spd)-525(matrix)-525(A)-525(and)-525(the)-525(right-hand)-525(side)-525(b)]TJ 0 -11.956 Td [(!)-525(using)-525(PSBLAS)-525(routines)-525(for)-525(sparse)-525(matrix)-525(/)-525(vector)-525(management)]TJ 0 -11.955 Td [(...)-525(...)]TJ 0 -11.955 Td [(!)]TJ 0 -11.955 Td [(!)-525(initialize)-525(the)-525(default)-525(multi-level)-525(preconditioner,)-525(i.e.)-525(V-cycle)]TJ 0 -11.955 Td [(!)-525(with)-525(basic)-525(smoothed)-525(aggregation,)-525(1)-525(hybrid)-525(forward/backward)]TJ 0 -11.956 Td [(!)-525(GS)-525(sweep)-525(as)-525(pre/post-smoother)-525(and)-525(UMFPACK)-525(as)-525(coarsest-level)]TJ 0 -11.955 Td [(!)-525(solver)]TJ 10.461 -11.955 Td [(call)-525(P%init\050P,'ML',info\051)]TJ -10.461 -11.955 Td [(!)]TJ 0 -11.955 Td [(!)-525(build)-525(the)-525(preconditioner)]TJ 10.461 -11.955 Td [(call)-525(P%hierarchy_build\050A,desc_A,P,info\051)]TJ 0 -11.956 Td [(call)-525(P%smoothers_build\050A,desc_A,P,info\051)]TJ -10.461 -23.91 Td [(!)]TJ 0 -11.955 Td [(!)-525(set)-525(the)-525(solver)-525(parameters)-525(and)-525(the)-525(initial)-525(guess)]TJ 10.461 -11.955 Td [(...)-525(...)]TJ -10.461 -11.955 Td [(!)]TJ 0 -11.956 Td [(!)-525(solve)-525(Ax=b)-525(with)-525(preconditioned)-525(CG)]TJ 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[(ilev)]TJ/F15 10.9091 Tf 22.909 0 Td [(.)]TJ ET q -1 0 0 1 891.361 556.702 cm +1 0 0 1 891.361 543.153 cm []0 d 0 J 0.398 w 0 0 m 0 13.549 l S Q q -1 0 0 1 292.625 556.502 cm +1 0 0 1 292.625 542.953 cm []0 d 0 J 0.398 w 0 0 m 598.736 0 l S Q 0 g 0 G BT -/F15 10.9091 Tf 423.488 525.207 Td [(T)83(able)-333(4:)-444(P)27(arameters)-333(de\014ning)-333(the)-334(aggregation)-333(algorithm)-333(\050con)28(tin)28(ued\051.)]TJ +/F15 10.9091 Tf 423.488 511.658 Td [(T)83(able)-333(4:)-444(P)27(arameters)-333(de\014ning)-333(the)-334(aggregation)-333(algorithm)-333(\050con)28(tin)28(ued\051.)]TJ 0 g 0 G 0 g 0 G ET @@ -4394,9 +4404,9 @@ Q endstream endobj -496 0 obj +493 0 obj << -/Length 10766 +/Length 9562 >> stream 0 g 0 G @@ -4415,318 +4425,285 @@ q 0 g 0 G 0 g 0 G q -1 0 0 1 0 209.219 cm +1 0 0 1 0 188.896 cm []0 d 0 J 0.398 w 0 0 m 602.12 0 l S Q q -1 0 0 1 0 195.471 cm +1 0 0 1 0 175.147 cm []0 d 0 J 0.398 w 0 0 m 0 13.549 l S Q 1 0 0 1 -299.826 -121.521 cm BT -/F44 10.9091 Tf 305.803 321.057 Td [(what)]TJ +/F44 10.9091 Tf 305.803 300.733 Td [(what)]TJ ET q -1 0 0 1 422.332 316.992 cm +1 0 0 1 422.332 296.668 cm []0 d 0 J 0.398 w 0 0 m 0 13.549 l S Q BT -/F41 10.9091 Tf 428.31 321.057 Td [(d)22(a)67(t)67(a)-378(type)]TJ +/F41 10.9091 Tf 428.31 300.733 Td [(d)22(a)67(t)67(a)-378(type)]TJ ET q -1 0 0 1 525.922 316.992 cm +1 0 0 1 525.922 296.668 cm []0 d 0 J 0.398 w 0 0 m 0 13.549 l S Q BT -/F44 10.9091 Tf 531.9 321.057 Td [(val)]TJ +/F44 10.9091 Tf 531.9 300.733 Td [(val)]TJ ET q -1 0 0 1 586.067 316.992 cm +1 0 0 1 586.067 296.668 cm []0 d 0 J 0.398 w 0 0 m 0 13.549 l S Q BT -/F41 10.9091 Tf 592.044 321.057 Td [(def)89(a)22(ul)67(t)]TJ +/F41 10.9091 Tf 592.044 300.733 Td [(def)89(a)22(ul)67(t)]TJ ET q -1 0 0 1 646.211 316.992 cm +1 0 0 1 646.211 296.668 cm []0 d 0 J 0.398 w 0 0 m 0 13.549 l S Q BT -/F41 10.9091 Tf 652.188 321.057 Td [(comments)]TJ +/F41 10.9091 Tf 652.188 300.733 Td [(comments)]TJ ET q -1 0 0 1 901.945 316.992 cm +1 0 0 1 901.945 296.668 cm []0 d 0 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\addcontentsline{toc}{section}{Abstract} -\textsc{MLD2P4 (Multi-Level Domain Decomposition Parallel Preconditioners Package based on -PSBLAS}) is a package of parallel algebraic multi-level preconditioners. -The first release made available various versions of -one-level additive and multi-level additive -and hybrid Schwarz preconditioners. -The package has been extended to include further multi-level cycles and smoothers widely used in -multigrid methods. -In the multi-level case, a purely algebraic approach -is applied to generate coarse-level corrections, so that no geometric background is needed -concerning the matrix to be preconditioned. The matrix is assumed to be square, real -or complex. +\textsc{MLD2P4 (Multi-Level Domain Decomposition Parallel Preconditioners Package +based on PSBLAS}) is a package of parallel algebraic multi-level preconditioners. +The first release of MLD2P4 made available multi-level additive and hybrid Schwarz +preconditioners, as well as one-level additive Schwarz preconditioners. The package +has been extended to include further multi-level cycles and smoothers widely used in +multigrid methods. In the multi-level case, a purely algebraic approach is applied to +generate coarse-level corrections, so that no geometric background is needed +concerning the matrix to be preconditioned. The matrix is assumed to be square, +real or complex. -MLD2P4 has been designed to provide scalable and easy-to-use preconditioners in the -context of the PSBLAS (Parallel Sparse Basic Linear Algebra Subprograms) +MLD2P4 has been designed to provide scalable and easy-to-use preconditioners +in the context of the PSBLAS (Parallel Sparse Basic Linear Algebra Subprograms) computational framework and can be used in conjuction with the Krylov solvers -available in this framework. MLD2P4 enables the user to easily specify different features -of an algebraic multi-level preconditioner, thus allowing to search +available in this framework. MLD2P4 enables the user to easily specify different +features of an algebraic multi-level preconditioner, thus allowing to search for the ``best'' preconditioner for the problem at hand. The package employs object-oriented design techniques in diff --git a/docs/src/background.tex b/docs/src/background.tex index 603a3e9d..8b600155 100644 --- a/docs/src/background.tex +++ b/docs/src/background.tex @@ -1,349 +1 @@ -\section{Multi-level Domain Decomposition Background\label{sec:background}} -\markboth{\textsc{MLD2P4 User's and Reference Guide}} - {\textsc{\ref{sec:background} Multi-level Domain Decomposition Background}} - -\emph{Domain Decomposition} (DD) preconditioners, coupled with Krylov iterative -solvers, are widely used in the parallel solution of large and sparse linear systems. -These preconditioners are based on the divide and conquer technique: the matrix -to be preconditioned is divided into submatrices, a ``local'' linear system -involving each submatrix is (approximately) solved, and the local solutions are used -to build a preconditioner for the whole original matrix. This process -often corresponds to dividing a physical domain associated to the original matrix -into subdomains, e.g. in a PDE discretization, to (approximately) solving the -subproblems corresponding to the subdomains and to building an approximate -solution of the original problem from the local solutions -\cite{Cai_Widlund_92,dd1_94,dd2_96}. - -\emph{Additive Schwarz} preconditioners are DD preconditioners using overlapping -submatrices, i.e.\ with some common rows, to couple the local information -related to the submatrices (see, e.g., \cite{dd2_96}). -The main motivation for choosing Additive Schwarz preconditioners is their -intrinsic parallelism. A drawback of these -preconditioners is that the number of iterations of the preconditioned solvers -generally grows with the number of submatrices. This may be a serious limitation -on parallel computers, since the number of submatrices usually matches the number -of available processors. Optimal convergence rates, i.e.\ iteration numbers -independent of the number of submatrices, can be obtained by correcting the -preconditioner through a suitable approximation of the original linear system -in a coarse space, which globally couples the information related to the single -submatrices. - -\emph{Two-level Schwarz} preconditioners are obtained -by combining basic (one-level) Sch\-warz preconditioners with a coarse-level -correction. In this context, the one-level preconditioner is often -called `smoother'. Different two-level preconditioners are obtained by varying the -choice of the smoother and of the coarse-level correction, and the -way they are combined \cite{dd2_96}. The same reasoning can be applied starting -from the coarse-level system, i.e.\ a coarse-space correction can be built -from this system, thus obtaining \emph{multi-level} preconditioners. - -It is worth noting that optimal preconditioners do not necessarily correspond -to minimum execution times. Indeed, to obtain effective multi-level preconditioners -a tradeoff between optimality of convergence and the cost of building and applying -the coarse-space corrections must be achieved. The choice of the number of levels, -i.e.\ of the coarse-space corrections, also affects the effectiveness of the -preconditioners. One more goal is to get convergence rates as less sensitive -as possible to variations in the matrix coefficients. - -Two main approaches can be used to build coarse-space corrections. The geometric approach -applies coarsening strategies based on the knowledge of some physical grid associated -to the matrix and requires the user to define grid transfer operators from the fine -to the coarse levels and vice versa. This may result difficult for complex geometries; -furthermore, suitable one-level preconditioners may be required to get efficient -interplay between fine and coarse levels, e.g.\ when matrices with highly varying coefficients -are considered. The algebraic approach builds coarse-space corrections using only matrix -information. It performs a fully automatic coarsening and enforces the interplay between -the fine and coarse levels by suitably choosing the coarse space and the coarse-to-fine -interpolation \cite{Stuben_01}. - -MLD2P4 uses a pure algebraic approach for building the sequence of coarse matrices -starting from the original matrix. The algebraic approach is based on the \emph{smoothed -aggregation} algorithm \cite{BREZINA_VANEK,VANEK_MANDEL_BREZINA}. A decoupled version -of this algorithm is implemented, where the smoothed aggregation is applied locally -to each submatrix \cite{TUMINARO_TONG}. In the next two subsections we provide -a brief description of the multi-level Schwarz preconditioners and of the smoothed -aggregation technique as implemented in MLD2P4. For further details the reader -is referred to \cite{para_04,aaecc_07,apnum_07,MLD2P4_TOMS,dd2_96}. - - -\subsection{Multi-level Schwarz Preconditioners\label{sec:multilevel}} - -The Multilevel preconditioners implemented in MLD2P4 are obtained by combining -AS preconditioners with coarse-space corrections; therefore -we first provide a sketch of the AS preconditioners. - -Given the linear system \Ref{system1}, -where $A=(a_{ij}) \in \Re^{n \times n}$ is a -nonsingular sparse matrix with a symmetric nonzero pattern, -let $G=(W,E)$ be the adjacency graph of $A$, where $W=\{1, 2, \ldots, n\}$ -and $E=\{(i,j) : a_{ij} \neq 0\}$ are the vertex set and the edge set of $G$, -respectively. Two vertices are called adjacent if there is an edge connecting -them. For any integer $\delta > 0$, a $\delta$-overlap -partition of $W$ can be defined recursively as follows. -Given a 0-overlap (or non-overlapping) partition of $W$, -i.e.\ a set of $m$ disjoint nonempty sets $W_i^0 \subset W$ such that -$\cup_{i=1}^m W_i^0 = W$, a $\delta$-overlap -partition of $W$ is obtained by considering the sets -$W_i^\delta \supset W_i^{\delta-1}$ obtained by including the vertices that -are adjacent to any vertex in $W_i^{\delta-1}$. - -Let $n_i^\delta$ be the size of $W_i^\delta$ and $R_i^{\delta} \in -\Re^{n_i^\delta \times n}$ the restriction operator that maps -a vector $v \in \Re^n$ onto the vector $v_i^{\delta} \in \Re^{n_i^\delta}$ -containing the components of $v$ corresponding to the vertices in -$W_i^\delta$. The transpose of $R_i^{\delta}$ is a -prolongation operator from $\Re^{n_i^\delta}$ to $\Re^n$. -The matrix $A_i^\delta=R_i^\delta A (R_i^\delta)^T \in -\Re^{n_i^\delta \times n_i^\delta}$ can be considered -as a restriction of $A$ corresponding to the set $W_i^{\delta}$. - -The \emph{classical one-level AS} preconditioner is defined by -\[ -M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T -(A_i^\delta)^{-1} R_i^{\delta}, -\] -where $A_i^\delta$ is assumed to be nonsingular. Its application -to a vector $v \in \Re^n$ within a Krylov solver requires the following -three steps: -\begin{enumerate} - \item restriction of $v$ as $v_i = R_i^{\delta} v$, $i=1,\ldots,m$; - \item solution of the linear systems $A_i^\delta w_i = v_i$, - $i=1,\ldots,m$; - \item prolongation and sum of the $w_i$'s, i.e. $w = \sum_{i=1}^m (R_i^{\delta})^T w_i$. -\end{enumerate} -Note that the linear systems at step 2 are usually solved approximately, -e.g.\ using incomplete LU factorizations such as ILU($p$), MILU($p$) and -ILU($p,t$) \cite[Chapter 10]{Saad_book}. - -A variant of the classical AS preconditioner that outperforms it -in terms of convergence rate and of computation and communication -time on parallel distributed-memory computers is the so-called \emph{Restricted AS -(RAS)} preconditioner~\cite{CAI_SARKIS,EFSTATHIOU}. It -is obtained by zeroing the components of $w_i$ corresponding to the -overlapping vertices when applying the prolongation. Therefore, -RAS differs from classical AS by the prolongation operators, -which are substituted by $(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$, -where $\tilde{R}_i^0$ is obtained by zeroing the rows of $R_i^\delta$ -corresponding to the vertices in $W_i^\delta \backslash W_i^0$: -\[ -M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T -(A_i^\delta)^{-1} R_i^{\delta}. -\] -Analogously, the AS variant called \emph{AS with Harmonic extension (ASH)} -is defined by -\[ M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T -(A_i^\delta)^{-1} \tilde{R}_i^0. -\] -We note that for $\delta=0$ the three variants of the AS preconditioner are -all equal to the block-Jacobi preconditioner. - -As already observed, the convergence rate of the one-level Schwarz -preconditioned iterative solvers deteriorates as the number $m$ of partitions -of $W$ increases \cite{dd1_94,dd2_96}. To reduce the dependency -of the number of iterations on the degree of parallelism we may -introduce a global coupling among the overlapping partitions by defining -a coarse-space approximation $A_C$ of the matrix $A$. -In a pure algebraic setting, $A_C$ is usually built with -the Galerkin approach. Given a set $W_C$ of \emph{coarse vertices}, -with size $n_C$, and a suitable restriction operator -$R_C \in \Re^{n_C \times n}$, $A_C$ is defined as -\[ -A_C=R_C A R_C^T -\] -and the coarse-level correction matrix to be combined with a generic -one-level AS preconditioner $M_{1L}$ is obtained as -\[ -M_{C}^{-1}= R_C^T A_C^{-1} R_C, -\] -where $A_C$ is assumed to be nonsingular. The application of $M_{C}^{-1}$ -to a vector $v$ corresponds to a restriction, a solution and -a prolongation step; the solution step, involving the matrix $A_C$, -may be carried out also approximately. - -The combination of $M_{C}$ and $M_{1L}$ may be -performed in either an additive or a multiplicative framework. -In the former case, the \emph{two-level additive} Schwarz preconditioner -is obtained: -\[ -M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}. -\] -Applying $M_{2L-A}^{-1}$ to a vector $v$ within a Krylov solver -corresponds to applying $M_{C}^{-1}$ -and $M_{1L}^{-1}$ to $v$ independently and then summing up -the results. - -In the multiplicative case, the combination can be -performed by first applying the smoother $M_{1L}^{-1}$ and then -the coarse-level correction operator $M_{C}^{-1}$: -\[ -\begin{array}{l} -w = M_{1L}^{-1} v, \\ -z = w + M_{C}^{-1} (v-Aw); -\end{array} -\] -this corresponds to the following \emph{two-level hybrid pre-smoothed} -Schwarz preconditioner: -\[ -M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}. -\] -On the other hand, by applying the smoother after the coarse-level correction, -i.e.\ by computing -\[ -\begin{array}{l} -w = M_{C}^{-1} v , \\ -z = w + M_{1L}^{-1} (v-Aw) , -\end{array} -\] -the \emph{two-level hybrid post-smoothed} -Schwarz preconditioner is obtained: -\[ -M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}. -\] -One more variant of two-level hybrid preconditioner is obtained by applying -the smoother before and after the coarse-level correction. In this case, the -preconditioner is symmetric if $A$, $M_{1L}$ and $M_{C}$ are symmetric. - -As previously noted, on parallel computers the number of submatrices usually matches -the number of available processors. When the size of the system to be preconditioned -is very large, the use of many processors, i.e.\ of many small submatrices, often -leads to a large coarse-level system, whose solution may be computationally expensive. -On the other hand, the use of few processors often leads to local sumatrices that -are too expensive to be processed on single processors, because of memory and/or -computing requirements. Therefore, it seems natural to use a recursive approach, -in which the coarse-level correction is re-applied starting from the current -coarse-level system. The corresponding preconditioners, called \emph{multi-level} -preconditioners, can significantly reduce the computational cost of preconditioning -with respect to the two-level case (see \cite[Chapter 3]{dd2_96}). -Additive and hybrid multilevel preconditioners -are obtained as direct extensions of the two-level counterparts. -For a detailed descrition of them, the reader is -referred to \cite[Chapter 3]{dd2_96}. -The algorithm for the application of a multi-level hybrid -post-smoothed preconditioner $M$ to a vector $v$, i.e.\ for the -computation of $w=M^{-1}v$, is reported, for -example, in Figure~\ref{fig:mlhpost_alg}. Here the number of levels -is denoted by $nlev$ and the levels are numbered in increasing order starting -from the finest one, i.e.\ the finest level is level 1; the coarse matrix -and the corresponding basic preconditioner at each level $l$ are denoted by $A_l$ and -$M_l$, respectively, with $A_1=A$, while the related restriction operator is -denoted by $R_l$. -% -\begin{figure}[t] -\begin{center} -\framebox{ -\begin{minipage}{.85\textwidth} {\small -\begin{tabbing} -\quad \=\quad \=\quad \=\quad \\[-1mm] -% -%! assign the finest matrix\\ -%$A_1 \leftarrow A$;\\[1mm] -%! define the number of levels $nlev$ \\[1mm] -%! define $nlev-1$ prolongators\\ -%$R_l^T, l=2, \ldots, nlev$;\\[1mm] -%! define $nlev-1$ coarser matrices\\ -%$A_l \leftarrow R_lA_{l-1}R_l^T, \; l=2, \ldots, nlev$;\\[1mm] -%! define the $nlev-1$ basic Schwarz preconditioners\\ -%$M_l$, basic preconditioner for $A_l \; l=1, \ldots, nlev-1$;\\[1mm] -%$! assign a vector $v$\\ -% -$v_1 = v$; \\[2mm] -\textbf{for $l=2, nlev$ do}\\[1mm] -\> ! transfer $v_{l-1}$ to the next coarser level\\ -\> $v_l = R_lv_{l-1}$ \\[1mm] -\textbf{endfor} \\[2mm] -! apply the coarsest-level correction\\[1mm] -$y_{nlev} = A_{nlev}^{-1} v_{nlev}$\\[2mm] -\textbf{for $l=nlev -1 , 1, -1$ do}\\[1mm] -\> ! transfer $y_{l+1}$ to the next finer level\\ -\> $y_l = R_{l+1}^T y_{l+1}$;\\[1mm] -\> ! compute the residual at the current level\\ -\> $r_l = v_l-A_l^{-1} y_l$;\\[1mm] -\> ! apply the basic Schwarz preconditioner to the residual\\ -\> $r_l = M_l^{-1} r_l$\\[1mm] -\> ! update $y_l$\\ -\> $y_l = y_l+r_l$\\ -\textbf{endfor} \\[1mm] -$w = y_1$; -\end{tabbing} -} -\end{minipage} -} -\caption{Application of the multi-level hybrid post-smoothed preconditioner.\label{fig:mlhpost_alg}} -\end{center} -\end{figure} -% - - -\subsection{Smoothed Aggregation\label{sec:aggregation}} - -In order to define the restriction operator $R_C$, which is used to compute -the coarse-level matrix $A_C$, MLD2P4 uses the \emph{smoothed aggregation} -algorithm described in \cite{BREZINA_VANEK,VANEK_MANDEL_BREZINA}. -The basic idea of this algorithm is to build a coarse set of vertices -$W_C$ by suitably grouping the vertices of $W$ into disjoint subsets -(aggregates), and to define the coarse-to-fine space transfer operator $R_C^T$ by -applying a suitable smoother to a simple piecewise constant -prolongation operator, to improve the quality of the coarse-space correction. - -Three main steps can be identified in the smoothed aggregation procedure: -\begin{enumerate} - \item coarsening of the vertex set $W$, to obtain $W_C$; - \item construction of the prolongator $R_C^T$; - \item application of $R_C$ and $R_C^T$ to build $A_C$. -\end{enumerate} -%\textbf{NOTA: Controllare cosa fa trilinos dopo il primo passo.} - -To perform the coarsening step, we have implemented the aggregation algorithm sketched -in \cite{apnum_07}. According to \cite{VANEK_MANDEL_BREZINA}, a modification of -this algorithm has been actually considered, -in which each aggregate $N_r$ is made of vertices of $W$ that are \emph{strongly coupled} -to a certain root vertex $r \in W$, i.e.\ -\[ N_r = \left\{s \in W: |a_{rs}| > \theta \sqrt{|a_{rr}a_{ss}|} \right\} - \cup \left\{ r \right\} , -\] -for a given $\theta \in [0,1]$. -Since this algorithm has a sequential nature, a \emph{decoupled} version of -it has been chosen, where each processor $i$ independently applies the algorithm to -the set of vertices $W_i^0$ assigned to it in the initial data distribution. This -version is embarrassingly parallel, since it does not require any data communication. -On the other hand, it may produce non-uniform aggregates near boundary vertices, -i.e.\ near vertices adjacent to vertices in other processors, and is strongly -dependent on the number of processors and on the initial partitioning of the matrix $A$. -Nevertheless, this algorithm has been chosen for the implementation in MLD2P4, -since it has been shown to produce good results in practice -\cite{aaecc_07,apnum_07,TUMINARO_TONG}. - -The prolongator $P_C=R_C^T$ is built starting from a \emph{tentative prolongator} -$P \in \Re^{n \times n_C}$, defined as -\begin{equation} -P=(p_{ij}), \quad p_{ij}= -\left\{ \begin{array}{ll} -1 & \quad \mbox{if} \; i \in V^j_C \\ -0 & \quad \mbox{otherwise} -\end{array} \right. . -\label{eq:tent_prol} -\end{equation} -$P_C$ is obtained by -applying to $P$ a smoother $S \in \Re^{n \times n}$: -\begin{equation} -P_C = S P, -\label{eq:smoothed_prol} -\end{equation} -in order to remove oscillatory components from the range of the prolongator -and hence to improve the convergence properties of the multi-level -Schwarz method \cite{BREZINA_VANEK,Stuben_01}. -A simple choice for $S$ is the damped Jacobi smoother: -\begin{equation} -S = I - \omega D^{-1} A , -\label{eq:jac_smoother} -\end{equation} -where the value of $\omega$ can be chosen -using some estimate of the spectral radius of $D^{-1}A$ \cite{BREZINA_VANEK}. -% -%\textbf{NOTA: filtering di $A$ nello smoothing, da implementare?} -% - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "userguide" -%%% End: +\section{Multigrid Background\label{sec:background}} \markboth{\textsc{MLD2P4 User's and Reference Guide}} {\textsc{\ref{sec:background} Multigrid Background}} Multigrid preconditioners, coupled with Krylov iterative solvers, are widely used in the parallel solution of large and sparse linear systems, because of their optimality in the solution of linear systems arising from the discretization of scalar elliptic Partial Differential Equations (PDEs) on regular grids. Optimality, also known as algorithmic scalability, is the property of having a computational cost per iteration that depends linearly on the problem size, and a convergence rate that is independent of the problem size. Multigrid preconditioners are based on a recursive application of a two-grid process consisting of smoother iterations and a coarse-space (or coarse-level) correction. The smoothers may be either basic iterative methods, such as the Jacobi and Gauss-Seidel ones, or more complex subspace-correction methods, such as the Schwarz ones. The coarse-space correction consists of solving, in an appropriately chosen coarse space, the residual equation associated with the approximate solution computed by the smoother, and of using the solution of this equation to correct the previous approximation. The transfer of information between the original (fine) space and the coarse one is performed by using suitable restriction and prolongation operators. The construction of the coarse space and the corresponding transfer operators is carried out by applying a so-called coarsening algorithm to the system matrix. Two main approaches can be used to perform coarsening: the geometric approach, which exploits the knowledge of some physical grid associated with the matrix and requires the user to define transfer operators from the fine to the coarse level and vice versa, and the algebraic approach, which builds the coarse-space correction and the associate transfer operators using only matrix information. The first approach may be difficult when the system comes from discretizations on complex geometries; furthermore, ad hoc one-level smoothers may be required to get an efficient interplay between fine and coarse levels, e.g., when matrices with highly varying coefficients are considered. The second approach performs a fully automatic coarsening and enforces the interplay between fine and coarse level by suitably choosing the coarse space and the coarse-to-fine interpolation (see, e.g., \cite{Briggs2000,Stuben_01,dd2_96} for details.) MLD2P4 uses a pure algebraic approach, based on the smoothed aggregation algorithm \cite{BREZINA_VANEK,VANEK_MANDEL_BREZINA}, for building the sequence of coarse matrices and transfer operators, starting from the original one. A decoupled version of this algorithm is implemented, where the smoothed aggregation is applied locally to each submatrix \cite{TUMINARO_TONG}. A brief description of the AMG preconditioners implemented in MLD2P4 is given in Sections~\ref{sec:multilevel}-\ref{sec:smoothers}. For further details the reader is referred to \cite{para_04,aaecc_07,apnum_07,MLD2P4_TOMS}. We note that optimal multigrid preconditioners do not necessarily correspond to minimum execution times in a parallel setting. Indeed, to obtain effective parallel multigrid preconditioners, a tradeoff between the optimality and the cost of building and applying the smoothers and the coarse-space corrections must be achieved. Effective parallel preconditioners require algorithmic scalability to be coupled with implementation scalability, i.e., a computational cost per iteration which remains (almost) constant as the number of parallel processors increases. \subsection{AMG preconditioners\label{sec:multilevel}} In order to describe the AMG preconditioners available in MLD2P4, we consider a linear system \begin{equation} Ax=b, \label{eq:system} \end{equation} where $A=(a_{ij}) \in \mathbb{R}^{n \times n}$ is a nonsingular sparse matrix; for ease of presentation we assume $A$ is real, but the results are valid for the complex case as well. Let us assume as finest index space the set of row (column) indices of $A$, i.e., $\Omega = \{1, 2, \ldots, n\}$. Any algebraic multilevel preconditioners implemented in MLD2P4 generates a hierarchy of index spaces and a corresponding hierarchy of matrices, \[ \Omega^1 \equiv \Omega \supset \Omega^2 \supset \ldots \supset \Omega^{nlev}, \quad A^1 \equiv A, A^2, \ldots, A^{nlev}, \] by using the information contained in $A$, without assuming any knowledge of the geometry of the problem from which $A$ originates. A vector space $\mathbb{R}^{n_{k}}$ is associated with $\Omega^k$, where $n_k$ is the size of $\Omega^k$. For all $k < nlev$, a restriction operator and a prolongation one are built, which connect two levels $k$ and $k+1$: $$ P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad R^k \in \mathbb{R}^{n_{k+1}\times n_k}; $$ %\[ % P^k: \mathbb{R}^{n_{k+1}} \longrightarrow \mathbb{R}^{n_k}, \quad % R^k: \mathbb{R}^{n_k} \longrightarrow \mathbb{R}^{n_{k+1}}; %\] the matrix $A^{k+1}$ is computed by using the previous operators according to the Galerkin approach, i.e., $$ A^{k+1}=R^kA^kP^k. $$ $R^k=(P^k)^T$ in the current implementation of MLD2P4. A smoother with iteration matrix $M^k$ is set up at each level $k < nlev$, and a solver is set up at the coarsest level, so that they are ready for application (for example, setting up a solver based on the $LU$ factorization means computing and storing the $L$ and $U$ factors). The construction of the hierachy of AMG components described so far corresponds to the so-called build phase of the preconditioner. \begin{figure}[t] \begin{center} \framebox{ \begin{minipage}{.85\textwidth} \begin{tabbing} \quad \=\quad \=\quad \=\quad \\[-3mm] procedure V-cycle$\left(k,A^k,b^k,u^k\right)$ \\[2mm] \>if $\left(k \ne nlev \right)$ then \\[1mm] \>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$ \\[1mm] \>\> $b^{k+1} = R^{k+1}\left(b^k - A^k u^k\right)$ \\[1mm] \>\> $u^{k+1} =$ V-cycle$\left(k+1,A^{k+1},b^{k+1},0\right)$ \\[1mm] \>\> $u^k = u^k + P^{k+1} u^{k+1}$ \\[1mm] \>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$ \\[1mm] \>else \\[1mm] \>\> $u^k = \left(A^k\right)^{-1} b^k$\\[1mm] \>endif \\[1mm] \>return $u^k$ \\[1mm] end \end{tabbing} \end{minipage} } \caption{Application phase of a V-cycle preconditioner.\label{fig:application_alg}} \end{center} \end{figure} The components produced in the build phase may be combined in several ways to obtain different multilevel preconditioners; this is done in the application phase, i.e., in the computation of a vector of type $w=B^{-1}v$, where $B$ denotes the preconditioner, usually within an iteration of a Krylov solver \cite{Saad_book}. An example of such a combination, known as V-cycle, is given in Figure~\ref{fig:application_alg}. In this case, a single iteration of the same smoother is used before and after the the recursive call to the V-cycle (i.e., in the pre-smoothing and post-smoothing phases); however, different choices can be performed. Other cycles can be defined; in MLD2P4, we implemented the standard V-cycle and W-cycle~\cite{Briggs2000}, and a version of the K-cycle described in~\cite{Notay2008}. \subsection{Smoothed Aggregation\label{sec:aggregation}} In order to define the prolongator $P^k$, used to compute the coarse-level matrix $A^{k+1}$, MLD2P4 uses the smoothed aggregation algorithm described in \cite{BREZINA_VANEK,VANEK_MANDEL_BREZINA}. The basic idea of this algorithm is to build a coarse set of indices $\Omega^{k+1}$ by suitably grouping the indices of $\Omega^k$ into disjoint subsets (aggregates), and to define the coarse-to-fine space transfer operator $P^k$ by applying a suitable smoother to a simple piecewise constant prolongation operator, with the aim of improving the quality of the coarse-space correction. Three main steps can be identified in the smoothed aggregation procedure: \begin{enumerate} \item aggregation of the indices set $\Omega^k$, to obtain $\Omega^{k+1}$; \item construction of the prolongator $P^k$; \item application of $P^k$ and $R^k=(P^k)^T$ to build $A^{k+1}$. \end{enumerate} In order to perform the coarsening step, the smoothed aggregation algorithm described in~\cite{VANEK_MANDEL_BREZINA} is used. In this algorithm, each index in $\Omega^{k+1}$ corresponds to an aggregate of $\Omega^k$, consisting of a suitably chosen index $j$ and of the indices $i$ that are strongly coupled to $j$, i.e., $$ |a_{ij}^k| > \theta \sqrt{|a_{ii}^ka_{jj}^k|}, $$ for a given $\theta \in [0,1]$. Since this algorithm has a sequential nature, a decoupled version of it is applied, where each processor $i$ independently executes the algorithm on the set of indices assigned to it in the initial data distribution. This version is embarrassingly parallel, since it does not require any data communication. On the other hand, it may produce some non-uniform aggregates and is strongly dependent on the number of processors and on the initial partitioning of the matrix $A$. Nevertheless, this parall algorithm has been chosen for MLD2P4, since it has been shown to produce good results in practice \cite{aaecc_07,apnum_07,TUMINARO_TONG}. The prolongator $P^k$ is built starting from a tentative prolongator $\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$, defined as $$ \bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k = \left\{ \begin{array}{ll} 1 & \quad \mbox{if} \; i \in \Omega^k_j, \\ 0 & \quad \mbox{otherwise}, \end{array} \right. \label{eq:tent_prol} $$ where $\Omega^k_j$ is the aggregate of $\Omega^k$ corresponding to the index $j \in \Omega^{k+1}$. $P^k$ is obtained by applying to $\bar{P}^k$ a smoother $S^k \in \mathbb{R}^{n_k \times n_k}$: $$ P^k = S^k \bar{P}^k, $$ in order to remove nonsmooth components from the range of the prolongator, and hence to improve the convergence properties of the multi-level method~\cite{BREZINA_VANEK,Stuben_01}. A simple choice for $S^k$ is the damped Jacobi smoother: $$ S^k = I - \omega^k (D^k)^{-1} A^k , $$ where $D^k$ is the diagonal matrix with the same diagonal entries as $A^k$, and $\omega^k$ is an approximation of $4/(3\rho^k)$, where $\rho^k$ is the spectral radius of $(D^k)^{-1}A^k$. computed by using some estimate of the spectral radius of $(D^k)^{-1}A^k$ \cite{BREZINA_VANEK}. \subsection{Smoothers and coarsest-level solvers\label{sec:smoothers}} The smoothers implemented in MLD2P4 include the Jacobi and block-Jacobi methods, a hybrid version of the forward and backward Gauss-Seidel methods, and the additive Schwarz (AS) ones (see, e.g., \cite{Saad_book,dd2_96}). The hybrid Gauss-Seidel version is considered because the original Gauss-Seidel method is inherently sequential. At each iteration of the hybrid version, each parallel process uses the most recent values of its own local variables and the values of the non-local variables computed at the previous iteration, obtained by exchanging data with other processes before the beginning of the current iteration. In the AS methods, the index space $\Omega^k$ is divided into $m_k$ subsets $\Omega^k_i$ of size $n_{k,i}$, possibly overlapping. For each $i$ we consider the restriction operator $R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$ % $R_i^k: \mathbb{R}^{n_k} \longrightarrow \mathbb{R}^{n_{k,i}}$ that maps a vector $x^k$ to the vector $x_i^k$ made of the components of $x^k$ with indices in $\Omega^k_i$, and the prolongation operator $P^k_i = (R_i^k)^T$. These operators are then used to build $A_i^k=R_i^kA^kP_i^k$, which is the restriction of $A^k$ to the index space $\Omega^k_i$. The classical AS preconditioner $M^k_{AS}$ is defined as \[ ( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k}, \] where $A_i^k$ is supposed to be nonsingular. We observe that an approximate inverse of $A_i^k$ is usually considered instead of $(A_i^k)^{-1}$. The setup of $S^k_{AS}$ during the multilevel build phase involves \begin{itemize} \item the definition of the index subspaces $\Omega_i^k$ and of the corresponding operators $R_i^k$ (and $P_i^k$); \item the computation of the submatrices $A_i^k$; \item the computation of their inverses (usually approximated through some form of incomplete factorization). \end{itemize} The computation of $z^k=M^k_{AS}w^k$, with $w^k \in \mathbb{R}^{n_k}$, during the multilevel application phase, requires \begin{itemize} \item the restriction of $w^k$ to the subspaces $\mathbb{R}^{n_{k,i}}$, i.e.\ $w_i^k = R_i^{k} w^k$; \item the computation of the vectors $z_i^k=(A_i^k)^{-1} w_i^k$; \item the prolongation and the sum of the previous vectors, i.e.\ $z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$. \end{itemize} Variants of the classical AS method, which use modifications of the restriction and prolongation operators, are also implemented in MLD2P4. Among them, the Restricted AS (RAS) preconditioner usually outperforms the classical AS preconditioner in terms of convergence rate and of computation and communication time on parallel distributed-memory computers, and is therefore the most widely used among the AS preconditioners~\cite{CAI_SARKIS}. Direct solvers based on sparse LU factorizations, implemented in the third party libraries reported in Section~\ref{sec:third_party}, can be applied as coarsest-level solvers by MLD2P4. Native inexact solvers based on incomplete LU factorizations, as well as Jacobi, hybrid (forward) Gauss-Seidel, and block Jacobi preconditioners are also available. Direct solvers usually lead to more effective preconditioners in terms of algorithmic scalability; however, this does not guarantee parallel efficiency. %%% Local Variables: %%% mode: latex %%% TeX-master: "userguide" %%% End: \ No newline at end of file diff --git a/docs/src/bibliography.tex b/docs/src/bibliography.tex index 2b5cb0b3..7d8290fa 100644 --- a/docs/src/bibliography.tex +++ b/docs/src/bibliography.tex @@ -1,229 +1,169 @@ -%\section{Bibliography\label{sec:bib}} -\begin{thebibliography}{99} -\addcontentsline{toc}{section}{\refname} -\markboth{\textsc{MLD2P4 User's and Reference Guide}} - {\textsc{References}} - -%\let\refname\relax - -% -%\bibitem{PARA04FOREST} -%G.~Bella, S.~Filippone, A.~De Maio, A., Testa, M.: -%A Simulation Model for Forest Fires. -%In: Dongarra, J., Madsen, K., Wasniewski, J. (eds.): -%Proceedings of PARA~04 Workshop on State of the Art -%in Scientific Computing. Lecture Notes in Computer Science, 3732. Berlin: -%Springer, 2005 -% -\bibitem{BREZINA_VANEK} -M.~Brezina, P.~Van{\v e}k, -{\em A Black-Box Iterative Solver Based on a Two-Level Schwarz Method}, -Computing, 63, 1999, 233--263. -% -\bibitem{para_04} -A.~Buttari, P.~D'Ambra, D.~di Serafino, S.~Filippone, -{\em Extending PSBLAS to Build Parallel Schwarz Preconditioners}, -in , J.~Dongarra, K.~Madsen, J.~Wasniewski, editors, -Proceedings of PARA~04 Workshop on State of the Art -in Scientific Computing, Lecture Notes in Computer Science, -Springer, 2005, 593--602. -% -\bibitem{aaecc_07} -A.~Buttari, P.~D'Ambra, D.~di~Serafino, S.~Filippone, -{\em 2LEV-D2P4: a package of high-performance preconditioners -for scientific and engineering applications}, -Applicable Algebra in Engineering, Communications and Computing, -18, 3, 2007, 223--239. -%Published online: 13 February 2007, {\tt http://dx.doi.org/10.1007/s00200-007-0035-z} -% -\bibitem{apnum_07} P.~D'Ambra, S.~Filippone, D.~di~Serafino, -{\em On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners}, -Applied Numerical Mathematics, Elsevier Science, -57, 11-12, 2007, 1181-1196. -%published online 3 February 2007, {\tt -% http://dx.doi.org/10.1016/j.apnum.2007.01.006} - -%% \bibitem{DOUGLAS} -%% R.E.~Bank and C.C.~Douglas, -%% {\em SMMP: Sparse Matrix Multiplication Package}, -%% Advances in Computational Mathematics, 1993, 1, 127-137. -%% (See also {\tt http://www.mgnet.org/~douglas/ccd-codes.html}) -% -% -%% \bibitem{CAI_SAAD} -%% X.~C.~Cai and Y.~Saad, -%% {\em Overlapping Domain Decomposition Algorithms for General Sparse Matrices}, -%% Numerical Linear Algebra with Applications, 3(3), pp.~221--237, 1996. -% -\bibitem{CAI_SARKIS} -X.~C.~Cai, M.~Sarkis, -{\em A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems}, -SIAM Journal on Scientific Computing, 21, 2, 1999, 792--797. -% -\bibitem{Cai_Widlund_92} -X.~C.~Cai, O.~B.~Widlund, -{\em Domain Decomposition Algorithms for Indefinite Elliptic Problems}, -SIAM Journal on Scientific and Statistical Computing, 13, 1, 1992, 243--258. -% -\bibitem{dd1_94} -T.~Chan and T.~Mathew, -{\em Domain Decomposition Algorithms}, -in A.~Iserles, editor, Acta Numerica 1994, 61--143. -Cambridge University Press. -% -\bibitem{MLD2P4_TOMS} -P.~D'Ambra, D.~di~Serafino, S.~Filippone, -\emph{MLD2P4: a Package of Parallel Multilevel -Algebraic Domain Decomposition Preconditioners -in Fortran 95}, ACM Trans. Math. Softw., 37(3), 2010. -% -\bibitem{UMFPACK} -T.A.~Davis, -{\em Algorithm 832: UMFPACK - an Unsymmetric-pattern Multifrontal -Method with a Column Pre-ordering Strategy}, -ACM Transactions on Mathematical Software, 30, 2004, 196--199. -(See also {\tt http://www.cise.ufl.edu/~davis/}) -% - -\bibitem{MUMPS} -P.R.~Amestoy, C.~Ashcraft, O.~Boiteau, A.~Buttari, J.~L'Excellent, C.~Weisbecker -{\em Improving multifrontal methods by means of block low-rank representations}, -SIAM SISC, volume 37, number 3, pages A1452-A1474. -(See also {\tt http://mumps.enseeiht.fr}) -% - -\bibitem{SUPERLU} -J.W.~Demmel, S.C.~Eisenstat, J.R.~Gilbert, X.S.~Li and J.W.H.~Liu, -A supernodal approach to sparse partial pivoting, -SIAM Journal on Matrix Analysis and Applications, 20, 3, 1999, 720--755. -% -\bibitem{blas3} -J.~J.~Dongarra, J.~Du Croz, I.~S.~Duff, S.~Hammarling, -\emph{A set of Level 3 Basic Linear Algebra Subprograms}, -ACM Transactions on Mathematical Software, 16, 1990, 1--17. -% -\bibitem{blas2} -J.~J.~Dongarra, J.~Du Croz, S.~Hammarling, R.~J.~Hanson, -\emph{An extended set of FORTRAN Basic Linear Algebra Subprograms}, -ACM Transactions on Mathematical Software, 14, 1988, 1--17. -% -\bibitem{BLACS} -J.~J.~Dongarra and R.~C.~Whaley, -{\em A User's Guide to the BLACS v.~1.1}, -Lapack Working Note 94, Tech.\ Rep.\ UT-CS-95-281, University of -Tennessee, March 1995 (updated May 1997). -% -%\bibitem{sblas_97} -%I.~Duff, M.~Marrone, G.~Radicati and C.~Vittoli, -%{\em Level 3 Basic Linear Algebra Subprograms for Sparse Matrices: -%a User Level Interface}, -%ACM Transactions on Mathematical Software, 23(3), pp.~379--401, 1997. -% -%\bibitem{sblas_02} -%I.~Duff, M.~Heroux and R.~Pozo, -%{\em An Overview of the Sparse Basic Linear -%Algebra Subprograms: the New Standard from the BLAS Technical Forum}, -%ACM Transactions on Mathematical Software, 28(2), pp.~239--267, 2002. -% -\bibitem{EFSTATHIOU} -E.~Efstathiou, J.~G.~Gander, -{\em Why Restricted Additive Schwarz Converges Faster than Additive Schwarz}, -BIT Numerical Mathematics, 43, 2003, 945--959. -% -\bibitem{PSBLASGUIDE} -S.~Filippone, A.~Buttari, -{\em PSBLAS-3.0 User's Guide. A Reference Guide for the Parallel Sparse BLAS Library}, 2012, -available from \texttt{http://www.ce.uniroma2.it/psblas/}. - -\bibitem{PSBLAS3} -Salvatore Filippone and Alfredo Buttari. -{\em {Object-Oriented Techniques for Sparse Matrix Computations in Fortran - 2003}.} -ACM Trans. on Math Software, 38(4), 2012. - -% -\bibitem{psblas_00} -S.~Filippone, M.~Colajanni, -{\em PSBLAS: A Library for Parallel Linear Algebra -Computation on Sparse Matrices}, -ACM Transactions on Mathematical Software, 26, 4, 2000, 527--550. -% -\bibitem{MPI2} -W.~Gropp, S.~Huss-Lederman, A.~Lumsdaine, E.~Lusk, B.~Nitzberg, W.~Saphir, M.~Snir, -{\em MPI: The Complete Reference. Volume 2 - The MPI-2 Extensions}, -MIT Press, 1998. -% -\bibitem{blas1} -C.~L.~Lawson, R.~J.~Hanson, D.~Kincaid, F.~T.~Krogh, -\emph{Basic Linear Algebra Subprograms for FORTRAN usage}, -ACM Transactions on Mathematical Software, 5, 1979, 308--323. -% -\bibitem{SUPERLUDIST} -X.~S.~Li, J.~W.~Demmel, {\em SuperLU\_DIST: A Scalable Distributed-memory -Sparse Direct Solver for Unsymmetric Linear Systems}, -ACM Transactions on Mathematical Software, 29, 2, 2003, 110--140. -% -%\bibitem{KIVA3PSBLAS} -%S.~Filippone, P.~D'Ambra, M.~Colajanni, -%{\em Using a Parallel Library of Sparse Linear Algebra in a Fluid Dynamics -%Applications Code on Linux Clusters}, -%in G.~Joubert, A.~Murli, F.~Peters, M.~Vanneschi, editors, -%Parallel Computing - Advances \& Current Issues, -%pp.~441--448, Imperial College Press, 2002. -% -%\bibitem{METIS} -%Karypis, G. and Kumar, V., -%{\em {METIS}: Unstructured Graph Partitioning and Sparse Matrix -% Ordering System}. -%Minneapolis, MN 55455: University of Minnesota, Department of -% Computer Science, 1995. -%Internet Address: {\verb|http://www.cs.umn.edu/~karypis|}. -%\bibitem{BLAS1} -%Lawson, C., Hanson, R., Kincaid, D. and Krogh, F., -% Basic {L}inear {A}lgebra {S}ubprograms for {F}ortran usage, -%{ACM Trans. Math. Softw.} vol.~{5}, 38--329, 1979. -% -%\bibitem{machiels} -%{Machiels, L. and Deville, M.} -%{\em Fortran 90: An entry to object-oriented programming for the solution -% of partial differential equations.} -%{ACM Trans. Math. Softw.} vol.~{23}, 32--49. -%\bibitem{metcalf} -%{Metcalf, M., Reid, J. and Cohen, M.} -%{\em Fortran 95/2003 explained.} -%{Oxford University Press}, 2004. -% -\bibitem{Saad_book} -Y.~Saad, -\emph{Iterative methods for sparse linear systems}, 2nd edition, -SIAM, 2003 - -\bibitem{dd2_96} -B.~Smith, P.~Bjorstad, W.~Gropp, -{\em Domain Decomposition: Parallel Multilevel Methods for Elliptic -Partial Differential Equations}, -Cambridge University Press, 1996. -% -\bibitem{MPI1} -M.~Snir, S.~Otto, S.~Huss-Lederman, D.~Walker, J.~Dongarra, -{\em MPI: The Complete Reference. Volume 1 - The MPI Core}, second edition, -MIT Press, 1998. -%% -\bibitem{Stuben_01} -K.~St\"{u}ben, -{\em An Introduction to Algebraic Multigrid}, -in A.~Sch\"{u}ller, U.~Trottenberg, C.~Oosterlee, Multigrid, -Academic Press, 2001. -% -\bibitem{TUMINARO_TONG} -R.~S.~Tuminaro, C.~Tong, -{\em Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines}, -in J. Donnelley, editor, Proceedings of SuperComputing 2000, Dallas, 2000. -% -\bibitem{VANEK_MANDEL_BREZINA} -P.~Van{\v e}k, J.~Mandel and M.~Brezina, -{\em Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems}, -Computing, 56, 1996, 179-196. -% - -\end{thebibliography} +%\section{Bibliography\label{sec:bib}} +\begin{thebibliography}{99} +\addcontentsline{toc}{section}{\refname} +\markboth{\textsc{MLD2P4 User's and Reference Guide}} + {\textsc{References}} + +%\let\refname\relax +% +\bibitem{BREZINA_VANEK} +M.~Brezina, P.~Van{\v e}k, +{\em A Black-Box Iterative Solver Based on a Two-Level Schwarz Method}, +Computing, 63, 1999, 233--263. +% +\bibitem{Briggs2000} +W.~L.~Briggs, V.~E.~Henson, S.~F.~ McCormick, +{\em A Multigrid Tutorial, Second Edition}, +SIAM, 2000. +% +\bibitem{para_04} +A.~Buttari, P.~D'Ambra, D.~di Serafino, S.~Filippone, +{\em Extending PSBLAS to Build Parallel Schwarz Preconditioners}, +in J.~Dongarra, K.~Madsen, J.~Wasniewski, editors, +Proceedings of PARA~04 Workshop on State of the Art +in Scientific Computing, Lecture Notes in Computer Science, +Springer, 2005, 593--602. +% +\bibitem{aaecc_07} +A.~Buttari, P.~D'Ambra, D.~di~Serafino, S.~Filippone, +{\em 2LEV-D2P4: a package of high-performance preconditioners +for scientific and engineering applications}, +Applicable Algebra in Engineering, Communications and Computing, +18 (3) 2007, 223--239. +%Published online: 13 February 2007, {\tt http://dx.doi.org/10.1007/s00200-007-0035-z} +% +\bibitem{apnum_07} P.~D'Ambra, S.~Filippone, D.~di~Serafino, +{\em On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners}, +Applied Numerical Mathematics, Elsevier Science, +57 (11-12), 2007, 1181-1196. +%published online 3 February 2007, {\tt +% http://dx.doi.org/10.1016/j.apnum.2007.01.006} +% +\bibitem{CAI_SARKIS} +X.~C.~Cai, M.~Sarkis, +{\em A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems}, +SIAM Journal on Scientific Computing, 21 (2), 1999, 792--797. +% +\bibitem{Cai_Widlund_92} +X.~C.~Cai, O.~B.~Widlund, +{\em Domain Decomposition Algorithms for Indefinite Elliptic Problems}, +SIAM Journal on Scientific and Statistical Computing, 13 (1), 1992, 243--258. +% +\bibitem{dd1_94} +T.~Chan and T.~Mathew, +{\em Domain Decomposition Algorithms}, +in A.~Iserles, editor, Acta Numerica 1994, 61--143. +Cambridge University Press. +% +\bibitem{MLD2P4_TOMS} +P.~D'Ambra, D.~di~Serafino, S.~Filippone, +\emph{MLD2P4: a Package of Parallel Multilevel +Algebraic Domain Decomposition Preconditioners +in Fortran 95}, ACM Trans. Math. Softw., 37(3), 2010, art. 30. +% +\bibitem{UMFPACK} +T.A.~Davis, +{\em Algorithm 832: UMFPACK - an Unsymmetric-pattern Multifrontal +Method with a Column Pre-ordering Strategy}, +ACM Transactions on Mathematical Software, 30, 2004, 196--199. +(See also {\tt http://www.cise.ufl.edu/~davis/}) +% +\bibitem{MUMPS} +P.R.~Amestoy, C.~Ashcraft, O.~Boiteau, A.~Buttari, J.~L'Excellent, C.~Weisbecker +{\em Improving multifrontal methods by means of block low-rank representations}, +SIAM Journal on Scientific Computing, volume 37 (3), 2015, A1452--A1474. +See also {\tt http://mumps.enseeiht.fr}. +% +\bibitem{SUPERLU} +J.W.~Demmel, S.C.~Eisenstat, J.R.~Gilbert, X.S.~Li and J.W.H.~Liu, +A supernodal approach to sparse partial pivoting, +SIAM Journal on Matrix Analysis and Applications, 20 (3), 1999, 720--755. +% +\bibitem{blas3} +J.~J.~Dongarra, J.~Du Croz, I.~S.~Duff, S.~Hammarling, +\emph{A set of Level 3 Basic Linear Algebra Subprograms}, +ACM Transactions on Mathematical Software, 16 (1) 1990, 1--17. +% +\bibitem{blas2} +J.~J.~Dongarra, J.~Du Croz, S.~Hammarling, R.~J.~Hanson, +\emph{An extended set of FORTRAN Basic Linear Algebra Subprograms}, +ACM Transactions on Mathematical Software, 14 (1) 1988, 1--17. +% +\bibitem{BLACS} +J.~J.~Dongarra and R.~C.~Whaley, +{\em A User's Guide to the BLACS v.~1.1}, +Lapack Working Note 94, Tech.\ Rep.\ UT-CS-95-281, University of +Tennessee, March 1995 (updated May 1997). +% +\bibitem{EFSTATHIOU} +E.~Efstathiou, J.~G.~Gander, +{\em Why Restricted Additive Schwarz Converges Faster than Additive Schwarz}, +BIT Numerical Mathematics, 43 (5), 2003, 945--959. +% +\bibitem{PSBLASGUIDE} +S.~Filippone, A.~Buttari, +{\em PSBLAS-3.0 User's Guide. A Reference Guide for the Parallel Sparse BLAS Library}, 2012, +available from \texttt{http://www.ce.uniroma2.it/psblas/}. +% +\bibitem{PSBLAS3} +Salvatore Filippone and Alfredo Buttari. +{\em Object-Oriented Techniques for Sparse Matrix Computations in Fortran 2003}. +ACM Transactions on on Mathematical Software, 38 (4), 2012, art. 23. +% +\bibitem{psblas_00} +S.~Filippone, M.~Colajanni, +{\em PSBLAS: A Library for Parallel Linear Algebra +Computation on Sparse Matrices}, +ACM Transactions on Mathematical Software, 26 (4), 2000, 527--550. +% +\bibitem{MPI2} +W.~Gropp, S.~Huss-Lederman, A.~Lumsdaine, E.~Lusk, B.~Nitzberg, W.~Saphir, M.~Snir, +{\em MPI: The Complete Reference. Volume 2 - The MPI-2 Extensions}, +MIT Press, 1998. +% +\bibitem{blas1} +C.~L.~Lawson, R.~J.~Hanson, D.~Kincaid, F.~T.~Krogh, +\emph{Basic Linear Algebra Subprograms for FORTRAN usage}, +ACM Transactions on Mathematical Software, 5 (3), 1979, 308--323. +% +\bibitem{SUPERLUDIST} +X.~S.~Li, J.~W.~Demmel, {\em SuperLU\_DIST: A Scalable Distributed-memory +Sparse Direct Solver for Unsymmetric Linear Systems}, +ACM Transactions on Mathematical Software, 29 (2), 2003, 110--140. +% +\bibitem{Notay2008} +Y.~Notay, P.~S.~Vassilevski, {\em Recursive Krylov-based multigrid cycles}, +Numerical Linear Algebra with Applications, 15 (5), 2008, 473--487. +% +\bibitem{Saad_book} +Y.~Saad, +{\em Iterative methods for sparse linear systems}, 2nd edition, SIAM, 2003. +% +\bibitem{dd2_96} +B.~Smith, P.~Bjorstad, W.~Gropp, +{\em Domain Decomposition: Parallel Multilevel Methods for Elliptic +Partial Differential Equations}, +Cambridge University Press, 1996. +% +\bibitem{MPI1} +M.~Snir, S.~Otto, S.~Huss-Lederman, D.~Walker, J.~Dongarra, +{\em MPI: The Complete Reference. Volume 1 - The MPI Core}, second edition, +MIT Press, 1998. +%% +\bibitem{Stuben_01} +K.~St\"{u}ben, +{\em An Introduction to Algebraic Multigrid}, +in A.~Sch\"{u}ller, U.~Trottenberg, C.~Oosterlee, Multigrid, +Academic Press, 2001. +% +\bibitem{TUMINARO_TONG} +R.~S.~Tuminaro, C.~Tong, +{\em Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines}, in J. Donnelley, editor, Proceedings of SuperComputing 2000, Dallas, 2000. +% +\bibitem{VANEK_MANDEL_BREZINA} +P.~Van{\v e}k, J.~Mandel and M.~Brezina, +{\em Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems}, +Computing, 56 (3) 1996, 179--196. +% + +\end{thebibliography} diff --git a/docs/src/building.tex b/docs/src/building.tex index 7a6a158e..b5596c1b 100644 --- a/docs/src/building.tex +++ b/docs/src/building.tex @@ -2,7 +2,7 @@ \markboth{\textsc{MLD2P4 User's and Reference Guide}} {\textsc{\ref{sec:building} Configuring and Building MLD2P4}} In order to build MLD2P4 it is necessary to set up a Makefile with appropriate -values for your system; this is done by means of the \verb|configure| +system-dependent variables; this is done by means of the \verb|configure| script. The distribution also includes the autoconf and automake sources employed to generate the script, but usually this is not needed to build the software. @@ -24,7 +24,7 @@ The following base libraries are needed: \item[BLAS] \cite{blas3,blas2,blas1} Many vendors provide optimized versions of BLAS; if no vendor version is available for a given platform, the ATLAS software - (\url{math-atlas.sourceforge.net/}) + (\url{math-atlas.sourceforge.net}) may be employed. The reference BLAS from Netlib (\url{www.netlib.org/blas}) are meant to define the standard behaviour of the BLAS interface, so they are not optimized for any @@ -35,14 +35,14 @@ The following base libraries are needed: libraries. Note that UMFPACK requires a full LAPACK library; our experience is that configuring ATLAS for building full LAPACK does not work in the correct way. Our advice is first to download the LAPACK tarfile from -\url{www.netlib.org/lapac} and install it independently of ATLAS. In this case, +\url{www.netlib.org/lapack} and install it independently of ATLAS. In this case, you need to modify the OPTS and NOOPT definitions for including -fPIC compilation option in the make.inc file of the LAPACK library. \item[MPI] \cite{MPI2,MPI1} A version of MPI is available on most high-performance computing systems. \item[PSBLAS] \cite{PSBLASGUIDE,psblas_00} Parallel Sparse BLAS (PSBLAS) is available from \url{www.ce.uniroma2.it/psblas}; version - 3.4.0 (or later) is required. Indeed, all the prerequisites + 3.5.0 (or later) is required. Indeed, all the prerequisites listed so far are also prerequisites of PSBLAS. \end{description} Please note that the four previous libraries must have Fortran @@ -50,7 +50,7 @@ interfaces compatible with MLD2P4; usually this means that they should all be built with the same compiler as MLD2P4. -\subsection{Optional third party libraries} +\subsection{Optional third party libraries\label{sec:third_party}} We provide interfaces to the following third-party software libraries; note that these are optional, but if you enable them some defaults @@ -61,11 +61,11 @@ for multi-level preconditioners may change to reflect their presence. A sparse LU factorization package included in the SuiteSparse library, available from \url{faculty.cse.tamu.edu/davis/suitesparse.html}; it provides sequential factorization and triangular system solution for double - precision real and complex data. We tested - version 4.5.4. Note that for configuring SuiteSparse you should provide the right -path to the BLAS and LAPACK libraries in the \verb|SuiteSparse_config/SuiteSparse_config.mk| file. + precision real and complex data. We tested version 4.5.4 of SuiteSparse. + Note that for configuring SuiteSparse you should provide the right path to the BLAS + and LAPACK libraries in the \verb|SuiteSparse_config/SuiteSparse_config.mk| file. \item[MUMPS] \cite{MUMPS} - A sparse LU factorization package available from \url{mumps.enseeiht.fr/}; + A sparse LU factorization package available from \url{mumps.enseeiht.fr}; it provides sequential and parallel factorizations and triangular system solution for single and double precision, real and complex data. We tested versions 4.10.0 and version 5.0.1. @@ -74,25 +74,24 @@ path to the BLAS and LAPACK libraries in the \verb|SuiteSparse_config/SuiteSpars \url{crd.lbl.gov/~xiaoye/SuperLU/}; it provides sequential factorization and triangular system solution for single and double precision, real and complex data. We tested version 4.3 and 5.0. If you installed BLAS from -ATLAS, remember to define the BLASLIB variable in the make.inc file. + ATLAS, remember to define the BLASLIB variable in the make.inc file. \item[SuperLU\_Dist] \cite{SUPERLUDIST} A sparse LU factorization package available from the same site as SuperLU; it provides parallel factorization and triangular system solution for double precision real and complex data. We tested version 3.3 and 4.2. If you installed BLAS from -ATLAS, remember to define the BLASLIB variable in the make.inc file and -to add the \verb|-std=c99| option to the C compiler options. -Note that this library requires the ParMETIS -library for parallel graph partitioning and fill-reducing matrix ordering available from -\url{glaros.dtc.umn.edu/gkhome/metis/parmetis/overview}. - + ATLAS, remember to define the BLASLIB variable in the make.inc file and + to add the \verb|-std=c99| option to the C compiler options. + Note that this library requires the ParMETIS + library for parallel graph partitioning and fill-reducing matrix ordering, available from + \url{glaros.dtc.umn.edu/gkhome/metis/parmetis/overview}. \end{description} -\subsection{Configuration options} -{\bf CONTROLLARE HELP DEL CONFIGURE: Versione MLD2P4, Versione PSBLAS, Influential Environmental Variables???} +\subsection{Configuration options} -To build MLD2P4 the first step is to use the \verb|configure| script -in the main directory to generate the necessary makefile(s). +In order to build MLD2P4, the first step is to use the \verb|configure| script +in the main directory to generate the necessary makefile. +%\textbf{Sono necessarie le parentesi intorno a s?} As a minimal example consider the following: \begin{verbatim} @@ -105,7 +104,7 @@ be specified with an {\em absolute} path). The full set of options may be looked at by issuing the command \verb|./configure --help|, which produces: \begin{verbatim} -`configure' configures MLD2P4 2.0 to adapt to many kinds of systems. +`configure' configures MLD2P4 2.1 to adapt to many kinds of systems. Usage: ./configure [OPTION]... [VAR=VALUE]... @@ -159,27 +158,55 @@ Fine tuning of the installation directories: --pdfdir=DIR pdf documentation [DOCDIR] --psdir=DIR ps documentation [DOCDIR] +Program names: + --program-prefix=PREFIX prepend PREFIX to installed program names + --program-suffix=SUFFIX append SUFFIX to installed program names + --program-transform-name=PROGRAM run sed PROGRAM on installed program names + Optional Features: --disable-option-checking ignore unrecognized --enable/--with options --disable-FEATURE do not include FEATURE (same as --enable-FEATURE=no) --enable-FEATURE[=ARG] include FEATURE [ARG=yes] + --disable-dependency-tracking speeds up one-time build + --enable-dependency-tracking do not reject slow dependency extractors --enable-serial Specify whether to enable a fake mpi library to run in serial mode. + --enable-long-integers Specify usage of 64 bits integers. Optional Packages: --with-PACKAGE[=ARG] use PACKAGE [ARG=yes] --without-PACKAGE do not use PACKAGE (same as --with-PACKAGE=no) --with-psblas=DIR The install directory for PSBLAS, for example, - --with-psblas=/opt/packages/psblas-3.3 + --with-psblas=/opt/packages/psblas-3.5 --with-psblas-incdir=DIR Specify the directory for PSBLAS includes. --with-psblas-libdir=DIR Specify the directory for PSBLAS library. + --with-ccopt additional CCOPT flags to be added: will prepend + to CCOPT + --with-fcopt additional FCOPT flags to be added: will prepend + to FCOPT + --with-libs List additional link flags here. For example, + --with-libs=-lspecial_system_lib or + --with-libs=-L/path/to/libs + --with-clibs additional CLIBS flags to be added: will prepend + to CLIBS + --with-flibs additional FLIBS flags to be added: will prepend + to FLIBS + --with-library-path additional LIBRARYPATH flags to be added: will + prepend to LIBRARYPATH + --with-include-path additional INCLUDEPATH flags to be added: will + prepend to INCLUDEPATH + --with-module-path additional MODULE_PATH flags to be added: will + prepend to MODULE_PATH --with-extra-libs List additional link flags here. For example, --with-extra-libs=-lspecial_system_lib or --with-extra-libs=-L/path/to/libs - --with-mumps=LIBNAME Specify the libname for MUMPS. Default: "-lsmumps - -ldmumps -lcmumps -lzmumps -lmumps_common -lpord" + --with-blas= use BLAS library + --with-blasdir= search for BLAS library in + --with-lapack= use LAPACK library + --with-mumps=LIBNAME Specify the libname for MUMPS. Default: autodetect + with minimum "-lmumps_common -lpord" --with-mumpsdir=DIR Specify the directory for MUMPS library and includes. Note: you will need to add auxiliary libraries with --extra-libs; this depends on how @@ -225,24 +252,22 @@ Some influential environment variables: CFLAGS C compiler flags CPPFLAGS C/C++/Objective C preprocessor flags, e.g. -I if you have headers in a nonstandard directory - CPP C preprocessor MPICC MPI C compiler command - F77 Fortran 77 compiler command - FFLAGS Fortran 77 compiler flags - MPIF77 MPI Fortran 77 compiler command MPIFC MPI Fortran compiler command + CPP C preprocessor Use these variables to override the choices made by `configure' or to help it to find libraries and programs with nonstandard names/locations. Report bugs to . \end{verbatim} -For instance, if a user has built and installed PSBLAS 3.4 under the + +For instance, if a user has built and installed PSBLAS 3.5 under the \verb|/opt| directory and is using the SuiteSparse package (which includes UMFPACK), then MLD2P4 might be configured with: \begin{verbatim} - ./configure --with-psblas=/opt/psblas-3.4/ \ + ./configure --with-psblas=/opt/psblas-3.5/ \ --with-umfpackincdir=/usr/include/suitesparse/ \end{verbatim} Once the configure script has completed execution, it will have @@ -253,7 +278,9 @@ install directory under the name \verb|Make.inc.MLD2P4|. To use the MUMPS solver package, the user has to add the appropriate options to the configure script; by default we are looking for the libraries -\verb|-ldmumps -lsmumps| \verb|-lzmumps -lzmumps -mumps_common -lpord|. +\verb|-ldmumps -lsmumps| \verb|-lzmumps -mumps_common -lpord|. +\textbf{Pasqua, c'era due volte lzmumps. L'ho eliminato, ma poi mi e' venuto +il dubbio che il secondo lzmumps dovesse essere modificato.} MUMPS often uses additional packages such as ScaLAPACK, ParMETIS, SCOTCH, as well as enabling OpenMP; in such cases it is necessary to add linker options with the \verb|--with-extra-libs| configure option. @@ -267,7 +294,7 @@ followed (optionally) by make install \end{verbatim} \subsection{Bug reporting} -If you find any bugs in our codes, please let us know at (DECIDERE A CHI FARE IL BUG REPORTING) +If you find any bugs in our codes, please let us know at \begin{rawhtml} \end{rawhtml} @@ -277,7 +304,8 @@ If you find any bugs in our codes, please let us know at (DECIDERE A CHI FARE IL \end{rawhtml} ; be aware that the amount of information needed to reproduce a problem in a parallel -program may vary quite a lot. +program may vary quite a lot. \textbf{A chi va fatto il bug reporting? La +mail inviata a questo indirizzo non viene mai letta.} \subsection{Example and test programs\label{sec:ex_and_test}} The package contains the \verb|examples| and \verb|tests| directories; both of them are further divided into \verb|fileread| and @@ -286,13 +314,14 @@ both of them are further divided into \verb|fileread| and \item[\tt examples] contains a set of simple example programs with a predefined choice of preconditioners, selectable via integer values. These are intended to get an acquaintance with the - multilevel preconditioners. + multi-level preconditioners available in MLD2P4. \item[\tt tests] contains a set of more sophisticated examples that will allow the user, via the input files in the \verb|runs| subdirectories, to experiment with the full range of preconditioners - implemented in the library. + implemented in the package. \end{description} The \verb|fileread| directories contain sample programs that read sparse matrices from files, according to the Matrix Market or the -Harwell-Boeing storage format; the \verb|pdegen| instead generate -matrices in full parallel mode from the discretization of a sample PDE. +Harwell-Boeing storage format; the \verb|pdegen| programs generate +matrices in full parallel mode from the discretization of a sample partial +differential equation. diff --git a/docs/src/gettingstarted.tex b/docs/src/gettingstarted.tex index ceacdce6..8f99a12c 100644 --- a/docs/src/gettingstarted.tex +++ b/docs/src/gettingstarted.tex @@ -41,9 +41,9 @@ The following steps are required: is multi-level, then two steps must be performed, as specified next. \begin{enumerate} \item[4.1] \emph{Build the aggregation hierarchy for a given matrix.} This is -performed by the routine \verb|hierarchy_bld|. +performed by the routine \verb|hierarchy_build|. \item[4.2] \emph{Build the preconditioner for a given matrix.} This is performed -by the routine \verb|smoothers_bld|. +by the routine \verb|smoothers_build|. \end{enumerate} If the selected preconditioner is one-level, it is built in a single step, performed by the routine \verb|bld|. @@ -118,7 +118,7 @@ on parallel computers. The code reported in Figure~\ref{fig:ex1} shows how to set and apply the default multi-level preconditioner available in the real double precision version of MLD2P4 (see Table~\ref{tab:precinit}). This preconditioner is chosen -by simply specifying \verb|'ML'| as second argument of \verb|P%init| +by simply specifying \verb|'ML'| as the second argument of \verb|P%init| (a call to \verb|P%set| is not needed) and is applied with the CG solver provided by PSBLAS (the matrix of the system to be solved is assumed to be positive definite). As previously observed, the modules @@ -179,8 +179,8 @@ the corresponding codes are available in \verb|examples/fileread/|. call P%init(P,'ML',info) ! ! build the preconditioner - call P%hierarchy_bld(A,desc_A,P,info) - call P%smoothers_bld(A,desc_A,P,info) + call P%hierarchy_build(A,desc_A,P,info) + call P%smoothers_build(A,desc_A,P,info) ! ! set the solver parameters and the initial guess @@ -264,8 +264,8 @@ boundary conditions are also available in the directory \verb|examples/pdegen|. call_P%set(P,'SMOOTHER_TYPE','BJAC',info) call P%set(P,'COARSE_SOLVE','BJAC',info) call P%set(P,'COARSE_SWEEPS',8,info) - call P%hierarchy_bld(A,desc_A,P,info) - call P%smoothers_bld(A,desc_A,P,info) + call P%hierarchy_build(A,desc_A,P,info) + call P%smoothers_build(A,desc_A,P,info) ... ... \end{verbatim} } @@ -291,8 +291,8 @@ boundary conditions are also available in the directory \verb|examples/pdegen|. call P%set('SMOOTHER_SWEEPS',2,info,pos='POST') call P%set('COARSE_SOLVE','MUMPS',info) call P%set('COARSE_MAT','DIST',info) - call P%hierarchy_bld(A,desc_A,P,info) - call P%smoothers_bld(A,desc_A,P,info) + call P%hierarchy_build(A,desc_A,P,info) + call P%smoothers_build(A,desc_A,P,info) ... ... ! solve Ax=b with preconditioned CG call psb_krylov('BICGSTAB',A,P,b,x,tol,desc_A,info) diff --git a/docs/src/license.tex b/docs/src/license.tex index dac694a3..46f2e4cb 100644 --- a/docs/src/license.tex +++ b/docs/src/license.tex @@ -13,13 +13,12 @@ terms: {\small (C) Copyright 2008, 2010, 2012, 2017 - Salvatore Filippone Cranfield University - Ambra Abdullahi Hassan University of Rome Tor Vergata - Alfredo Buttari CNRS-IRIT, Toulouse - Pasqua D'Ambra ICAR-CNR, Naples - Daniela di Serafino Second University of Naples + Salvatore Filippone Cranfield University, Cranfield, UK + Ambra Abdullahi Hassan University of Rome Tor Vergata, Rome, IT + Alfredo Buttari CNRS-IRIT, Toulouse, FR + Pasqua D'Ambra IAC-CNR, Naples, IT + Daniela di Serafino University of Campania L. Vanvitelli, Caserta, IT - Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: diff --git a/docs/src/overview.tex b/docs/src/overview.tex index c89d78d9..30079b4b 100644 --- a/docs/src/overview.tex +++ b/docs/src/overview.tex @@ -3,10 +3,9 @@ {\textsc{\ref{sec:overview} General Overview}} The \textsc{Multi-Level Domain Decomposition Parallel Preconditioners Package based on -PSBLAS (MLD2P4}) provides parallel Algebraic MultiGrid (AMG) and domain decomposition -preconditioners, designed to provide scalable and easy-to-use preconditioners -multi-level Schwarz preconditioners~\cite{Stuben_01,dd2_96}, -to be used in the iterative solutions of sparse linear systems: +PSBLAS (MLD2P4}) provides parallel Algebraic MultiGrid (AMG) and Domain +Decomposition preconditioners (see, e.g., \cite{Briggs2000,Stuben_01,dd2_96}), +to be used in the iterative solution of linear systems, \begin{equation} Ax=b, \label{system1} @@ -17,22 +16,34 @@ where $A$ is a square, real or complex, sparse matrix. %Dovremmo implementare uno smoothed prolongator %adeguato e fare qualcosa di consistente anche con 1-lev Schwarz.} % -Multi-level preconditioners can be obtained by combining several AMG cycles (V, W, K) with -different smoothers (Jacobi, hybrid forward/backward Gauss-Seidel, block-Jacobi, additive Schwarz methods). -An algebraic approach is used to -generate a hierarchy of coarse-level matrices and operators, without -explicitly using any information on the geometry of the original problem, e.g., -the discretization of a PDE. The smoothed aggregation technique is applied -as algebraic coarsening strategy~\cite{BREZINA_VANEK,VANEK_MANDEL_BREZINA}. -Either exact or approximate solvers are available to solve the coarsest-level system. Specifically, -different versions of sparse LU factorizations from external packages, and native incomplete -LU factorizations and iterative block-Jacobi solvers can be used. -All smoothers can be also exploited as one-level preconditioners. +The name of the package comes from its original implementation, containing +multi-level additive and hybrid Schwarz preconditioners, as well as one-level additive +Schwarz preconditioners. The current version extends the original plan by including +multi-level cycles and smoothers widely used in multigrid methods. + +The multi-level preconditioners implemented in MLD2P4 are obtained by combining +AMG cycles with smoothers and coarsest-level solvers. The V-, W-, and +K-cycles~\cite{Briggs2000,Notay2008} are available, which allow to define +almost all the preconditioners in the package, including the multi-level hybrid +Schwarz ones; a specific cycle is implemented to obained multi-level additive +Schwarz preconditioners. The Jacobi, hybrid +%\footnote{see Note 2 in Table~\ref{tab:p_coarse}, p.~28.} +forward/backward Gauss-Seidel, block-Jacobi, and additive Schwarz methods +are available as smoothers. An algebraic approach is used to generate a hierarchy of +coarse-level matrices and operators, without explicitly using any information on the +geometry of the original problem, e.g., the discretization of a PDE. To this end, +the smoothed aggregation technique~\cite{BREZINA_VANEK,VANEK_MANDEL_BREZINA} +is applied. Either exact or approximate solvers can be used on the coarsest-level +system. Specifically, different sparse LU factorizations from external +packages, and native incomplete LU factorizations and Jacobi, hybrid Gauss-Seidel, +and block-Jacobi solvers are available. All smoothers can be also exploited as one-level +preconditioners. MLD2P4 is written in Fortran~2003, following an object-oriented design through the exploitation of features -such as abstract data type creation, functional overloading, and -dynamic memory management. +such as abstract data type creation, type extension, functional overloading, and +dynamic memory management. % \textbf{Va bene cos\'{i} o \`e meglio +% fare riferimento alle classi?} The parallel implementation is based on a Single Program Multiple Data (SPMD) paradigm. Single and double precision implementations of MLD2P4 are available for both the @@ -40,53 +51,53 @@ real and the complex case, which can be used through a single interface. MLD2P4 has been designed to implement scalable and easy-to-use -multilevel preconditioners in the context of the PSBLAS -(Parallel Sparse BLAS) computational framework~\cite{psblas_00,PSBLAS3}. -PSBLAS provides basic linear algebra +multilevel preconditioners in the context of the PSBLAS (Parallel Sparse BLAS) +computational framework~\cite{psblas_00,PSBLAS3}. PSBLAS provides basic linear algebra operators and data management facilities for distributed sparse matrices, -as well as parallel Krylov solvers which can be coupled with the MLD2P4 preconditioners. +as well as parallel Krylov solvers which can be used with the MLD2P4 preconditioners. The choice of PSBLAS has been mainly motivated by the need of having a portable and efficient software infrastructure implementing ``de facto'' standard parallel sparse linear algebra kernels, to pursue goals such as performance, portability, modularity ed extensibility in the development of the preconditioner package. On the other hand, the implementation of MLD2P4 has led to some revisions and extentions of the original PSBLAS kernels. -The inter-process comunication required -by MLD2P4 is encapsulated into the PSBLAS routines, except few cases where -MPI~\cite{MPI1} is explicitly called \textbf{\'E ancora cosi???}. Therefore, MLD2P4 can be run on any parallel -machine where PSBLAS and MPI implementations are available. +The inter-process comunication required by MLD2P4 is encapsulated +in the PSBLAS routines;% , except few cases where MPI~\cite{MPI1} is explicitly called. +% \textbf{E' ancora cos\'{i} o adesso \`e tutto incapsulato in PSBLAS?} +therefore, MLD2P4 can be run on any parallel machine where PSBLAS +implementations are available. -MLD2P4 has a layered and modular software architecture where three main layers can be identified. -The lower layer consists of the PSBLAS kernels, the middle one implements +MLD2P4 has a layered and modular software architecture where three main layers can be +identified. The lower layer consists of the PSBLAS kernels, the middle one implements the construction and application phases of the preconditioners, and the upper one provides a uniform interface to all the preconditioners. This architecture allows for different levels of use of the package: -few black-box routines at the upper layer allow non-expert users to easily -build any preconditioner available in MLD2P4 and to apply it within a PSBLAS Krylov solver; -{\bf facilities are also available that allow more expert users to extend the set of smoothers -and solvers for building new versions of preconditioners.} +few black-box routines at the upper layer allow all users to easily +build and apply any preconditioner available in MLD2P4; +facilities are also available allowing expert users to extend the set of smoothers +and solvers for building new versions of the preconditioners (see +Section~\ref{sec:adding}). -We note that the user interface of MLD2P4 2.1 ({\bf Perche 2.1 e non 2.0???...Ricordarsi di cambiare il configure}) -has been extended with respect to the previous versions -in order to separate the construction -of the multi-level hierarchy from the construction of the smoothers and solvers, and to allow for more flexibility -at each level. -The software architecture described in~\cite{MLD2P4_TOMS} has significantly evolved too, in order to fully exploit the -Fortran~2003 features implemented in PSBLAS 3. +We note that the user interface of MLD2P4 2.1 has been extended with respect to the +previous versions in order to separate the construction of the multi-level hierarchy from +the construction of the smoothers and solvers, and to allow for more flexibility +at each level. The software architecture described in~\cite{MLD2P4_TOMS} has significantly +evolved too, in order to fully exploit the Fortran~2003 features implemented in PSBLAS 3. However, compatibility with previous versions has been preserved. -This guide is organized as follows. General information on the distribution of the source code -is reported in Section~\ref{sec:distribution}, while details on the configuration -and installation of the package are given in Section~\ref{sec:building}. A short description of -the preconditioners implemented in MLD2P4 is provided -in Section~\ref{sec:background}, to help the users in choosing among them. -The basics for building and applying the preconditioners -with the Krylov solvers implemented in PSBLAS are reported in Section~\ref{sec:started}, where the -Fortran codes of a few sample programs are also shown. A reference guide for -the upper-layer routines of MLD2P4, that are the user interface, is provided -in Section~\ref{sec:userinterface}. The error handling mechanism used by the package is briefly described -in Section~\ref{sec:errors}. The copyright terms concerning the distribution and modification -of MLD2P4 are reported in Appendix~\ref{sec:license}. +This guide is organized as follows. General information on the distribution of the source +code is reported in Section~\ref{sec:distribution}, while details on the configuration +and installation of the package are given in Section~\ref{sec:building}. A short description +of the preconditioners implemented in MLD2P4 is provided in Section~\ref{sec:background}, +to help the users in choosing among them. The basics for building and applying the +preconditioners with the Krylov solvers implemented in PSBLAS are reported +in~Section~\ref{sec:started}, where the Fortran codes of a few sample programs +are also shown. A reference guide for the user interface routines is provided +in Section~\ref{sec:userinterface}. Information on the extension of the package +through the addition of new smoothers and solvers is reported in Section~\ref{sec:adding}. +The error handling mechanism used by the package +is briefly described in Section~\ref{sec:errors}. The copyright terms concerning the +distribution and modification of MLD2P4 are reported in Appendix~\ref{sec:license}. %%% Local Variables: %%% mode: latex diff --git a/docs/src/userguide.tex b/docs/src/userguide.tex index db7a7762..07cb2688 100644 --- a/docs/src/userguide.tex +++ b/docs/src/userguide.tex @@ -154,7 +154,6 @@ based on PSBLAS} \include{overview} \include{distribution} \include{building} - \include{background} \include{gettingstarted} \include{userinterface} @@ -162,7 +161,7 @@ based on PSBLAS} \clearpage \appendix \include{license} -\cleardoublepage +\clearpage \include{bibliography} \end{document} diff --git a/docs/src/userinterface.tex b/docs/src/userinterface.tex index cd796aec..6e982e95 100644 --- a/docs/src/userinterface.tex +++ b/docs/src/userinterface.tex @@ -4,7 +4,7 @@ The basic user interface of MLD2P4 consists of eight routines. The six routines \verb|init|, \verb|set|, -\verb|hierarchy_bld|, \verb|smoothers_bld|, +\verb|hierarchy_build|, \verb|smoothers_build|, \verb|bld|, and \verb|apply| encapsulate all the functionalities for the setup and the application of any multi-level and one-level preconditioner implemented in the package. @@ -199,10 +199,9 @@ coarsest-level solvers, and shortcuts are available in this case too (see Table~\ref{tab:p_coarse}). \\ \textbf{Remark 3.} In general, a coarsest-level solver cannot be used with -both the replicated and distributed coarsest-matrix layout, and vice versa; -therefore, setting the solver after the layout may change the layout, and setting -the layout after the solver may change the solver, if the choices of the two -parameters do not agree. +both the replicated and distributed coarsest-matrix layout; +therefore, setting the solver after the layout may change the layout. +Similarly, setting the layout after the solver may change the solver. More precisely, UMFPACK and SuperLU require the coarsest-level matrix to be replicated, while SuperLU\_Dist requires it to be distributed. @@ -368,7 +367,9 @@ of levels. } \\ & How the damping parameter $\omega$ in the smoothed aggregation is obtained: either via an estimate of the spectral radius of - $D^{-1}A$, or explicily + $D^{-1}A$, where $A$ is the matrix at the current + level and $D$ is the diagonal matrix with + the same diagonal entires as $A$, or explicily specified by the user. \\ \hline \verb|mld_aggr_eig_| \par \verb|AGGR_EIG| & \verb|character(len=*)| & \texttt{'A\_NORMI'} @@ -420,13 +421,13 @@ the parameter \texttt{ilev}.} \\ & \texttt{'MUMPS'} \par \texttt{'UMF'} \par \texttt{'SLU'} \par \texttt{'SLUDIST'} \par \texttt{'JACOBI'} \par \texttt{'GS'} \par \texttt{'BJAC'} - & See~Note~1 + & See~Note. & Solver used at the coarsest level: sequential LU from MUMPS, UMFPACK, or SuperLU (plus tri\-an\-gular solve); distributed LU from MUMPS or SuperLU\_Dist (plus triangular solve); - point-Jacobi, hybrid Gauss-Seidel (see Note~2) or block-Jacobi. \par + point-Jacobi, hybrid Gauss-Seidel or block-Jacobi. \par Note that \texttt{UMF} and \texttt{SLU} require the coarsest matrix to be replicated, \texttt{SLUDIST}, \texttt{JACOBI}, \texttt{GS} and \texttt{BJAC} require it to be @@ -440,7 +441,7 @@ the parameter \texttt{ilev}.} \\ \verb|mld_coarse_subsolve_| \par \verb|COARSE_SUBSOLVE| & \verb|character(len=*)| & \texttt{'ILU'} \par \texttt{'ILUT'} \par \texttt{'MILU'} \par \texttt{'MUMPS'} \par \texttt{'SLU'} \par \texttt{'UMF'} - & See~Note~1 + & See~Note. & Solver for the diagonal blocks of the coarse matrix, in case the block Jacobi solver is chosen as coarsest-level solver: ILU($p$), ILU($p,t$), @@ -449,7 +450,7 @@ the parameter \texttt{ilev}.} \\ Note that UMFPACK and SuperLU\_Dist are available only in double precision. \\ \hline -\multicolumn{5}{|l|}{{\bfseries Note 1.} Defaults for \texttt{mld\_coarse\_solve\_} and +\multicolumn{5}{|l|}{{\bfseries Note.} Defaults for \texttt{mld\_coarse\_solve\_} and \texttt{mld\_coarse\_subsolve\_} are chosen in the following order:} \\ \multicolumn{5}{|l|}{single precision version -- \texttt{MUMPS} if installed, then \texttt{SLU} if installed, @@ -457,11 +458,6 @@ the parameter \texttt{ilev}.} \\ \multicolumn{5}{|l|}{double precision version -- \texttt{UMF} if installed, then \texttt{MUMPS} if installed, then \texttt{SLU} if installed, \texttt{ILU} otherwise.}\\ -\multicolumn{5}{|l|}{{\bfseries Note 2.} The hybrid Gauss-Seidel method is -between the Gauss-Seidel and Jacobi methods: at each iteration, the process-} \\ -\multicolumn{5}{|l|}{es use the most recent values of their own local variables, and the values of -the non-local variables computed at the previ-}\\ -\multicolumn{5}{|l|}{ous iteration.}\\ \hline \end{tabular} \end{center} @@ -512,7 +508,7 @@ level (continued).\label{tab:p_coarse_1}} & Type of smoother used in the multi-level preconditioner: point-Jacobi, hybrid (forward) Gauss-Seidel, hybrid backward Gauss-Seidel, block-Jacobi, and - Additive Schwarz. See Note for details on hybrix Gauss-Seidel.\par + Additive Schwarz. \par It is ignored by one-level preconditioners. \\ \hline \verb|mld_sub_solve_| \par \verb|SUB_SOLVE| & \verb|character(len=*)| & \texttt{'JACOBI'} \par @@ -541,11 +537,7 @@ level (continued).\label{tab:p_coarse_1}} \verb|mld_sub_ovr_| \par \verb|SUB_OVR| & \verb|integer| & Any integer \par number~$\ge 0$ & 1 - & Number of overlap layers, for Additive Schwarz only. \\ \hline -\multicolumn{5}{|l|}{{\bfseries Note.} The hybrid Gauss-Seidel method is -between the Gauss-Seidel and Jacobi methods: at each iteration, the processes use the} \\ -\multicolumn{5}{|l|}{most recent values of their own local variables, and the values of -the non-local variables computed at the previous iteration.}\\ + & Number of overlap layers, for Additive Schwarz only. \\ \hline \end{tabular} \end{center} @@ -565,13 +557,17 @@ the non-local variables computed at the previous iteration.}\\ & \texttt{'HALO'} & Type of restriction operator, for Additive Schwarz only: \texttt{HALO} for taking into account the overlap, \texttt{NONE} - for neglecting it. \\ \hline + for neglecting it. \par + Note that \texttt{HALO} must be chosen for + the classical Addditive Schwarz smoother and its RAS variant.\\ \hline \verb|mld_sub_prol_| \par \verb|SUB_PROL| & \verb|character(len=*)| & \texttt{'SUM'} \par \texttt{'NONE'} & \texttt{'NONE'} & Type of prolongation operator, for Additive Schwarz only: \texttt{SUM} for adding the contributions from the overlap, \texttt{NONE} - for neglecting them. \\ \hline + for neglecting them. \par + Note that \texttt{SUM} must be chosen for the classical Additive + Schwarz smoother, and \texttt{NONE} for its RAS variant. \\ \hline \verb|mld_sub_fillin_| \par \verb|SUB_FILLIN| & \verb|integer| & Any integer \par number~$\ge 0$ & 0 @@ -601,16 +597,16 @@ the non-local variables computed at the previous iteration.}\\ \clearpage -\subsection{Subroutine bld\label{sec:precbld}} +\subsection{Subroutine build\label{sec:precbld}} \begin{center} -\verb|call p%bld(a,desc_a,info)|\\ +\verb|call p%build(a,desc_a,info)|\\ \end{center} \noindent This routine builds the one-level preconditioner \verb|p| according to the requirements made by the user through the routines \verb|init| and \verb|set| -(see Sections~\ref{sec:hier_bld} and~\ref{sec:smoothers_bld} for multi-level preconditioners). +(see Sections~\ref{sec:hier_bld} and~\ref{sec:smooth_bld} for multi-level preconditioners). {\vskip1.5\baselineskip\noindent\large\bfseries Arguments} \smallskip @@ -643,10 +639,10 @@ In this case, the routine can be used to build multi-level preconditioners too. \clearpage -\subsection{Subroutine hierarchy\_bld\label{sec:hier_bld}} +\subsection{Subroutine hierarchy\_build\label{sec:hier_bld}} \begin{center} -\verb|call p%hierarchy_bld(a,desc_a,info)|\\ +\verb|call p%hierarchy_build(a,desc_a,info)|\\ \end{center} \noindent @@ -676,18 +672,18 @@ single/double precision version of MLD2P4 under use. \clearpage -\subsection{Subroutine smoothers\_bld\label{sec:smoothers_bld}} +\subsection{Subroutine smoothers\_build\label{sec:smooth_bld}} \begin{center} -\verb|call p%smoothers_bld(a,desc_a,p,info)|\\ +\verb|call p%smoothers_build(a,desc_a,p,info)|\\ \end{center} \noindent This routine builds the smoothers and the coarsest-level solvers for the multi-level preconditioner \verb|p|, according to the requirements made by the user through the routines \verb|init| and \verb|set|, and based on the aggregation -hierarchy produced by a previous call to \verb|hierarchy_bld| +hierarchy produced by a previous call to \verb|hierarchy_build| (see Section~\ref{sec:hier_bld}). {\vskip1.5\baselineskip\noindent\large\bfseries Arguments} \smallskip @@ -804,8 +800,8 @@ as follows: \noindent This routine prints a description of the preconditioner \verb|p| to the standard output or -to a file. It must be called after \verb|hierachy_bld| and \verb|smoothers_bld|, -or \verb|bld|, have been called. +to a file. It must be called after \verb|hierachy_build| and \verb|smoothers_build|, +or \verb|build|, have been called. {\vskip1.5\baselineskip\noindent\large\bfseries Arguments} \smallskip