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diff --git a/docs/html/img91.png b/docs/html/img91.png index efb2a515..4a631dd6 100644 Binary files a/docs/html/img91.png and b/docs/html/img91.png differ diff --git a/docs/html/img92.png b/docs/html/img92.png index b69a0964..ed89fe01 100644 Binary files a/docs/html/img92.png and b/docs/html/img92.png differ diff --git a/docs/html/img93.png b/docs/html/img93.png index c0282ff0..925d3460 100644 Binary files a/docs/html/img93.png and b/docs/html/img93.png differ diff --git a/docs/html/index.html b/docs/html/index.html index f026ce9d..1f0d2ee7 100644 --- a/docs/html/index.html +++ b/docs/html/index.html @@ -1,6 +1,6 @@ - + userhtml @@ -9,7 +9,7 @@ - + @@ -69,17 +69,15 @@ based on PSBLAS


-Software version: 2.1 +Software version: 2.2
-July 31, 2017 +July 31, 2018

- - - +


diff --git a/docs/html/node1.html b/docs/html/node1.html index 202c3887..fc552ff5 100644 --- a/docs/html/node1.html +++ b/docs/html/node1.html @@ -1,6 +1,6 @@ - + Abstract @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node10.html b/docs/html/node10.html index a880f838..37ec11f4 100644 --- a/docs/html/node10.html +++ b/docs/html/node10.html @@ -1,6 +1,6 @@ - + Bug reporting @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node11.html b/docs/html/node11.html index 7b4e7b1b..13e572cb 100644 --- a/docs/html/node11.html +++ b/docs/html/node11.html @@ -1,6 +1,6 @@ - + Example and test programs @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node12.html b/docs/html/node12.html index b2da8f7e..0a40de2f 100644 --- a/docs/html/node12.html +++ b/docs/html/node12.html @@ -1,6 +1,6 @@ - + Multigrid Background @@ -9,7 +9,7 @@ - + @@ -88,19 +88,19 @@ are considered. The second approach performs a fully automatic coarsening and en interplay between fine and coarse level by suitably choosing the coarse space and the coarse-to-fine interpolation (see, e.g., [3,23,21] for details.) + HREF="node36.html#Stuben_01">24,22] for details.)

MLD2P4 uses a pure algebraic approach, based on the smoothed aggregation algorithm [2,25], + HREF="node36.html#VANEK_MANDEL_BREZINA">26], 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 [24]. + HREF="node36.html#TUMINARO_TONG">25]. 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 [ - + AMG preconditioners @@ -9,7 +9,7 @@ - + @@ -67,7 +67,7 @@ Ax=b,
\begin{displaymath} Ax=b, @@ -80,7 +80,7 @@ where <!-- MATH $A=(a_{ij}) \in \mathbb{R}^{n \times n}$ --> <SPAN CLASS=$A=(a_{ij}) \in \mathbb{R}^{n \times n}$ is a nonsingular sparse matrix; for ease of presentation we assume $\Omega = \{1, 2, \ldots, n\}$. Any algebraic multilevel preconditioners implemented in MLD2P4 generates @@ -116,7 +116,8 @@ a hierarchy of index spaces and a corresponding hierarchy of matrices, \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} + 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}">

@@ -132,28 +133,28 @@ 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 $k+1$: @@ -167,14 +168,17 @@ P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad --> \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} + 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}">

the matrix $A^{k+1}$ is computed by using the previous operators according to the Galerkin approach, i.e., @@ -188,9 +192,11 @@ A^{k+1}=R^kA^kP^k. --> \begin{displaymath} A^{k+1}=R^kA^kP^k. \end{displaymath} + ALT="\begin{displaymath} +A^{k+1}=R^kA^kP^k. +\end{displaymath}">

@@ -199,10 +205,10 @@ In the current implementation of MLD2P4 we have $R^k=(P^k)^T$ 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 @@ -211,7 +217,7 @@ is set up at the coarsest level, so that they are ready for application SRC="img19.png" ALT="$LU$"> factorization means computing and storing the $L$ and \framebox{ \begin{minipage}{.85\textwidth} \begin{tabbing} \quad \=\quad \=\quad... -...mm] \>endif [1mm] \>return $u^k$ [1mm] end \end{tabbing} \end{minipage} } + ALT="\framebox{ +\begin{minipage}{.85\textwidth} +\begin{tabbing} +\quad \=\quad \=\quad... +...[1mm] +\>endif [1mm] +\>return $u^k$ [1mm] +end +\end{tabbing}\end{minipage}}">
@@ -263,14 +276,14 @@ end 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 [20]. An example of such a combination, known as + HREF="node36.html#Saad_book">21]. An example of such a combination, known as V-cycle, is given in Figure 1. 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 @@ -278,7 +291,7 @@ performed. Other cycles can be defined; in MLD2P4, we implemented the standard V and W-cycle [3], and a version of the K-cycle described in [19]. + HREF="node36.html#Notay2008">20].

diff --git a/docs/html/node14.html b/docs/html/node14.html index a4d6105f..4a6e3560 100644 --- a/docs/html/node14.html +++ b/docs/html/node14.html @@ -1,6 +1,6 @@ - + Smoothed Aggregation @@ -9,7 +9,7 @@ - + @@ -54,27 +54,27 @@ Smoothed 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 [2,25]. + HREF="node36.html#VANEK_MANDEL_BREZINA">26]. 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. @@ -84,26 +84,26 @@ prolongation operator, with the aim of improving the quality of the coarse-space

  1. aggregation of the indices of $\Omega^k$ to obtain $\Omega^{k+1}$;
  2. construction of the prolongator $P^k$;
  3. application of $P^k$ and $R^k=(P^k)^T$ to build $A^{k+1}$.
    4. @@ -111,18 +111,18 @@ prolongation operator, with the aim of improving the quality of the coarse-space

    In order to perform the coarsening step, the smoothed aggregation algorithm described in [25] is used. In this algorithm, + HREF="node36.html#VANEK_MANDEL_BREZINA">26] is used. In this algorithm, each index $j \in \Omega^{k+1}$ corresponds to an aggregate $\Omega^k_j$ of $\Omega^k$, consisting of a suitably chosen index $\theta \in [0,1]$ (see [25] for the details). + HREF="node36.html#VANEK_MANDEL_BREZINA">26] for the details). Since this algorithm has a sequential nature, a decoupled version of it is applied, where each processor independently executes the algorithm on the set of indices assigned to it in the initial data @@ -180,11 +182,11 @@ MLD2P4, since it has been shown to produce good results in practice [5,7,24]. + HREF="node36.html#TUMINARO_TONG">25].

    The prolongator $P^k$ is built starting from a tentative prolongator $j \in \Omega^{k+1}$. $P^k$ is obtained by applying to $\bar{P}^k$ a smoother \begin{displaymath} P^k = S^k \bar{P}^k, \end{displaymath} + ALT="\begin{displaymath} +P^k = S^k \bar{P}^k, +\end{displaymath}">

    @@ -267,9 +275,9 @@ in order to remove nonsmooth components from the range of the prolongator, and hence to improve the convergence properties of the multilevel method [2,23]. + HREF="node36.html#Stuben_01">24]. A simple choice for $S^k$ is the damped Jacobi smoother: @@ -282,24 +290,26 @@ S^k = I - \omega^k (D^k)^{-1} A^k_F , --> \begin{displaymath} S^k = I - \omega^k (D^k)^{-1} A^k_F , \end{displaymath} + ALT="\begin{displaymath} +S^k = I - \omega^k (D^k)^{-1} A^k_F , +\end{displaymath}">

    where $D^k$ is the diagonal matrix with the same diagonal entries as $A^k$, $A^k_F = (\bar{a}_{ij}^k)$ is the filtered matrix defined as @@ -321,17 +331,20 @@ where +\bar{a}_{ij}^k = +\left \{ \begin{array}{ll} +a_{ij}^k & \m... +...ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k), +\end{displaymath}">
    \begin{displaymath} -\bar{a}_{ij}^k = \left \{ \begin{array}{ll} a_{ij}^k & ... -...ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k), \end{displaymath} (5)

    and $\omega^k$ is an approximation of $\rho^k$. Note that for systems coming from uniformly elliptic problems, filtering the matrix $A^k$ has little or no effect, and $A^k$ can be used instead of $A^k_F$. The latter choice is the default in MLD2P4. diff --git a/docs/html/node15.html b/docs/html/node15.html index bc801420..db064fff 100644 --- a/docs/html/node15.html +++ b/docs/html/node15.html @@ -1,6 +1,6 @@ - + Smoothers and coarsest-level solvers @@ -9,7 +9,7 @@ - + @@ -55,8 +55,8 @@ Smoothers and coarsest-level solvers 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., [20,21]). + HREF="node36.html#Saad_book">21,22]).

    The hybrid Gauss-Seidel @@ -68,7 +68,7 @@ the beginning of the current iteration.

    In the AS methods, the index space $\Omega^k$ is divided into $\Omega^k_i$ of size $n_{k,i}$, possibly overlapping. For each $i$ we consider the restriction operator \begin{displaymath} ( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k}, \end{displaymath} + ALT="\begin{displaymath} +( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k}, +\end{displaymath}">

    @@ -170,7 +172,7 @@ involves SRC="img62.png" ALT="$\Omega_i^k$">     and of the corresponding operators $R_i^k$ (and
    • the restriction of $w^k$ to the subspaces $\mathbb{R}^{n_{k,i}}$, i.e. + Getting Started @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node17.html b/docs/html/node17.html index 830f4f6c..bb8015ed 100644 --- a/docs/html/node17.html +++ b/docs/html/node17.html @@ -1,6 +1,6 @@ - + Examples @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node18.html b/docs/html/node18.html index fc22110c..614a5f09 100644 --- a/docs/html/node18.html +++ b/docs/html/node18.html @@ -1,6 +1,6 @@ - + User Interface @@ -9,7 +9,7 @@ - + @@ -83,14 +83,14 @@ i.e., matrix data structure;
    • the arrays containing the vectors $v$ and $w$ involved in the preconditioner application $w=B^{-1}v$ must be of type psb_xvect_type with x = diff --git a/docs/html/node19.html b/docs/html/node19.html index 1ea569b5..70549f3d 100644 --- a/docs/html/node19.html +++ b/docs/html/node19.html @@ -1,6 +1,6 @@ - + Method init @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node2.html b/docs/html/node2.html index 27a15212..18abc9b0 100644 --- a/docs/html/node2.html +++ b/docs/html/node2.html @@ -1,6 +1,6 @@ - + Contents @@ -9,7 +9,7 @@ - + @@ -122,9 +122,7 @@ Contents

      - - - +


      diff --git a/docs/html/node20.html b/docs/html/node20.html index 6d8135bf..c688ab59 100644 --- a/docs/html/node20.html +++ b/docs/html/node20.html @@ -1,6 +1,6 @@ - + Method set @@ -9,7 +9,7 @@ - + @@ -247,7 +247,7 @@ solver is changed to the default sequential solver.


      -
      +
      @@ -300,7 +300,7 @@ number

      -
      +
      Table 2: Parameters defining the multilevel cycle and the number of cycles to @@ -285,7 +285,7 @@ Note that hybrid Multiplicative Schwarz is equivalent to V-cycle and
      Any integer

      number $\ge 1$

      		 1
      @@ -344,7 +344,7 @@ Parameters defining the aggregation algorithm. @@ -358,7 +358,7 @@ Parameters defining the aggregation algorithm. @@ -376,18 +376,22 @@ Currently, only the SYMDEC option applies decoupled aggregation to the sparsity pattern of $A+A^T$. - - - + + + @@ -415,7 +419,7 @@ of levels.


      -
      +
      Table 3: Parameters defining the aggregation algorithm. @@ -319,17 +319,17 @@ Parameters defining the aggregation algorithm.
      Any number

      $> 0$

      		 $\lfloor 40 \sqrt[3]{n} \rfloor$, where $n$ is the dimension of the matrix at the finest levelAny number

      $> 1$

      		 1.5Any integer

      number $> 1$

      		 20
      'AGGR_TYPE' character(len=*)'VMB''VMB'Type of aggregation algorithm: currently, the scalar aggregation - algorithm by Vanek, Mandel and Brezina is implemented - [25].'SOC1''SOC1', + 'SOC2'Type of aggregation algorithm: currently, + we implement to measures of strength of + connection, the one by Vanek, Mandel + and Brezina [26], + and the one by Gratton et al [16].
      'AGGR_PROL' character(len=*)
      +


      -
      +
      Table 4: Parameters defining the aggregation algorithm (continued). @@ -440,18 +444,17 @@ Parameters defining the aggregation algorithm (continued). descending degrees of the nodes in the matrix graph. -
      -'AGGR_THRESH'
      'AGGR_THRESH' real(kind_parameter) Any real

      number $\in [0, 1]$

      		 0.01 The threshold $\theta$ in the aggregation algorithm, see (3) in Section 4.2. @@ -468,7 +471,7 @@ number 
      Note. Different thresholds at different levels, such as those used in [25, Section 5.1], can be easily set by + HREF="node36.html#VANEK_MANDEL_BREZINA">26, Section 5.1], can be easily set by invoking the rou-
      tine set with @@ -483,7 +486,7 @@ the parameter ilev.
      Table 5: Parameters defining the coarse-space correction at the coarsest @@ -590,7 +593,7 @@ Note that UMF and SLU require the coarsest


      -
      +
      @@ -621,7 +624,7 @@ number Any integer

      number $\ge 0$

      @@ -635,12 +638,12 @@ number Any real

      number $\ge 0$

      Table 6: Parameters defining the coarse-space correction at the coarsest @@ -609,7 +612,7 @@ level (continued).
      Any integer

      number $> 0$

      		 1000 Drop tolerance $t$ in the ILU(

      -
      +
      @@ -767,7 +770,7 @@ Parameters defining the smoother or the details of the one-level preconditioner. @@ -783,7 +786,7 @@ Parameters defining the smoother or the details of the one-level preconditioner.


      -
      +
      Table 7: Parameters defining the smoother or the details of the one-level preconditioner. @@ -748,7 +751,7 @@ Parameters defining the smoother or the details of the one-level preconditioner.
      Any integer

      number $\ge 0$

      		Any integer

      number $\ge 0$

      @@ -856,22 +859,21 @@ Parameters defining the smoother or the details of the one-level preconditioner - + diff --git a/docs/html/node21.html b/docs/html/node21.html index 8a1d324e..be357628 100644 --- a/docs/html/node21.html +++ b/docs/html/node21.html @@ -1,6 +1,6 @@ - +Method hierarchy_build @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node22.html b/docs/html/node22.html index 0bb9ab16..d9fa00f9 100644 --- a/docs/html/node22.html +++ b/docs/html/node22.html @@ -1,6 +1,6 @@ - +Method smoothers_build @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node23.html b/docs/html/node23.html index 445ce6f3..54375c9a 100644 --- a/docs/html/node23.html +++ b/docs/html/node23.html @@ -1,6 +1,6 @@ - +Method build @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node24.html b/docs/html/node24.html index be5f014a..357893fa 100644 --- a/docs/html/node24.html +++ b/docs/html/node24.html @@ -1,6 +1,6 @@ - +Method apply @@ -9,7 +9,7 @@ - + @@ -62,9 +62,9 @@ This method computes $y = op(B^{-1}) x$, where $y = op(B^{-1})\, x$, where $B$ is a previously built @@ -91,7 +91,7 @@ and hence it is completely transparent to the user. diff --git a/docs/html/node25.html b/docs/html/node25.html index da626cdc..23211355 100644 --- a/docs/html/node25.html +++ b/docs/html/node25.html @@ -1,6 +1,6 @@ - +Method free @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node26.html b/docs/html/node26.html index 494e7369..f07bcf9f 100644 --- a/docs/html/node26.html +++ b/docs/html/node26.html @@ -1,6 +1,6 @@ - +Method descr @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node27.html b/docs/html/node27.html index b4165d36..64b9a5db 100644 --- a/docs/html/node27.html +++ b/docs/html/node27.html @@ -1,6 +1,6 @@ - +Auxiliary Methods @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node28.html b/docs/html/node28.html index ffbacc9b..813e8e25 100644 --- a/docs/html/node28.html +++ b/docs/html/node28.html @@ -1,6 +1,6 @@ - +Method: dump @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node29.html b/docs/html/node29.html index 888861cf..da717d1c 100644 --- a/docs/html/node29.html +++ b/docs/html/node29.html @@ -1,6 +1,6 @@ - +Method: clone @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node3.html b/docs/html/node3.html index a5e870ba..02b8228b 100644 --- a/docs/html/node3.html +++ b/docs/html/node3.html @@ -1,6 +1,6 @@ - +General Overview @@ -9,7 +9,7 @@ - + @@ -57,8 +57,8 @@ General Overview PSBLAS (MLD2P4) provides parallel Algebraic MultiGrid (AMG) and Domain Decomposition preconditioners (see, e.g., [3,23,21]), + HREF="node36.html#Stuben_01">24,22]), to be used in the iterative solution of linear systems,
      @@ -72,7 +72,7 @@ Ax=b,
      Table 8: Parameters defining the smoother or the details of the one-level preconditioner @@ -840,7 +843,7 @@ Parameters defining the smoother or the details of the one-level preconditioner
      Any integer

      number $\ge 0$

      		real(kind_parameter) Any real number $\ge 0$ 0 Drop tolerance $t$ in the ILU($p,t$) factorization.
      -
      The local part of the vector $x$. Note that type and kind_parameter must be chosen according @@ -137,28 +137,28 @@ and hence it is completely transparent to the user. $op(B^{-1}) = B^{-1}$ --> $op(B^{-1}) = B^{-1}$; if trans = 'T','t' then $op(B^{-1}) = B^{-T}$ (transpose of $B^{-1})$; if trans = 'C','c' then $op(B^{-1}) = B^{-C}$ (conjugate transpose of $B^{-1})$.
      \begin{displaymath} Ax=b, @@ -95,7 +95,7 @@ multilevel cycles and smoothers widely used in multigrid methods. AMG cycles with smoothers and coarsest-level solvers. The V-, W-, and K-cycles [<A HREF=3,19] are available, which allow to define + HREF="node36.html#Notay2008">20] are available, which allow to define almost all the preconditioners in the package, including the multilevel hybrid Schwarz ones; a specific cycle is implemented to obtain multilevel additive Schwarz preconditioners. The Jacobi, hybrid @@ -105,7 +105,7 @@ coarse-level matrices and operators, without explicitly using any information on geometry of the original problem, e.g., the discretization of a PDE. To this end, the smoothed aggregation technique [2,25] + HREF="node36.html#VANEK_MANDEL_BREZINA">26] 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, @@ -116,8 +116,7 @@ preconditioners. MLD2P4 is written in Fortran 2003, following an object-oriented design through the exploitation of features 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 +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 diff --git a/docs/html/node30.html b/docs/html/node30.html index e5cfa8fa..873b2304 100644 --- a/docs/html/node30.html +++ b/docs/html/node30.html @@ -1,6 +1,6 @@ - + Method: sizeof @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node31.html b/docs/html/node31.html index a2044f73..1243f28c 100644 --- a/docs/html/node31.html +++ b/docs/html/node31.html @@ -1,6 +1,6 @@ - + Method: allocate_wrk @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node32.html b/docs/html/node32.html index 66687d38..5805fd18 100644 --- a/docs/html/node32.html +++ b/docs/html/node32.html @@ -1,6 +1,6 @@ - + Method: free_wrk @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node33.html b/docs/html/node33.html index a9e120a5..463a9879 100644 --- a/docs/html/node33.html +++ b/docs/html/node33.html @@ -1,6 +1,6 @@ - + Adding new smoother and solver objects to MLD2P4 @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node34.html b/docs/html/node34.html index 150cb8e1..a32c73cd 100644 --- a/docs/html/node34.html +++ b/docs/html/node34.html @@ -1,6 +1,6 @@ - + Error Handling @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node35.html b/docs/html/node35.html index 5a7991e0..369bfe3a 100644 --- a/docs/html/node35.html +++ b/docs/html/node35.html @@ -1,6 +1,6 @@ - + License @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node36.html b/docs/html/node36.html index bf1feef1..0504c1f0 100644 --- a/docs/html/node36.html +++ b/docs/html/node36.html @@ -1,6 +1,6 @@ - + Bibliography @@ -9,7 +9,7 @@ - + @@ -138,53 +138,60 @@ 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. -

      16 +

      16 +
      +S. Gratton, P. Henon, P. Jiranek and X. Vasseur, + Reducing complexity of algebraic multigrid by aggregation, +Numerical Lin. Algebra with Applications, 2016, 23:501-518 + +

      +

      17
      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. -

      17 +

      18
      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. -

      18 +

      19
      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. -

      19 +

      20
      Y. Notay, P. S. Vassilevski, Recursive Krylov-based multigrid cycles, Numerical Linear Algebra with Applications, 15 (5), 2008, 473-487. -

      20 +

      21
      Y. Saad, Iterative methods for sparse linear systems, 2nd edition, SIAM, 2003. -

      21 +

      22
      B. Smith, P. Bjorstad, W. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, 1996. -

      22 +

      23
      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. -

      23 +

      24
      K. Stüben, An Introduction to Algebraic Multigrid, in A. Schüller, U. Trottenberg, C. Oosterlee, Multigrid, Academic Press, 2001. -

      24 +

      25
      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. -

      25 +

      26
      P. Vanek, J. Mandel, M. Brezina, Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems, diff --git a/docs/html/node37.html b/docs/html/node37.html index 0eb0b5e6..657869fa 100644 --- a/docs/html/node37.html +++ b/docs/html/node37.html @@ -1,6 +1,6 @@ - + About this document ... @@ -9,7 +9,7 @@ - + @@ -47,7 +47,7 @@ About this document ...

      This document was generated using the -LaTeX2HTML translator Version 2017.2 (Released Jan 23, 2017) +LaTeX2HTML translator Version 2018 (Released Feb 1, 2018)

      Copyright © 1993, 1994, 1995, 1996, Nikos Drakos, @@ -60,7 +60,7 @@ Mathematics Department, Macquarie University, Sydney. The command line arguments were:
      latex2html -local_icons -noaddress -dir ../../html userhtml.tex

      -The translation was initiated on 2018-01-27 +The translation was initiated on 2018-05-14

      diff --git a/docs/html/node4.html b/docs/html/node4.html index 7b2ac12c..c478e331 100644 --- a/docs/html/node4.html +++ b/docs/html/node4.html @@ -1,6 +1,6 @@ - + Code Distribution @@ -9,7 +9,7 @@ - + @@ -80,7 +80,7 @@ constant --> \begin{displaymath}\verb\vert mld_version_string_\vert\end{displaymath} diff --git a/docs/html/node5.html b/docs/html/node5.html index 3e5ba8d6..a15ea834 100644 --- a/docs/html/node5.html +++ b/docs/html/node5.html @@ -1,6 +1,6 @@ - + Contributors @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node6.html b/docs/html/node6.html index 8c177d8c..54be0f4b 100644 --- a/docs/html/node6.html +++ b/docs/html/node6.html @@ -1,6 +1,6 @@ - + Configuring and Building MLD2P4 @@ -9,7 +9,7 @@ - + diff --git a/docs/html/node7.html b/docs/html/node7.html index 13445741..7eed7f21 100644 --- a/docs/html/node7.html +++ b/docs/html/node7.html @@ -1,6 +1,6 @@ - + Prerequisites @@ -9,7 +9,7 @@ - + @@ -59,7 +59,7 @@ Prerequisites
      [11,12,17] Many vendors provide optimized versions + HREF="node36.html#blas1">18] Many vendors provide optimized versions of BLAS; if no vendor version is available for a given platform, the ATLAS software (
      MPI
      [16,22] A version of MPI is available on most + HREF="node36.html#MPI2">17,23] A version of MPI is available on most high-performance computing systems.
      diff --git a/docs/html/node8.html b/docs/html/node8.html index f1b81928..b008d3ae 100644 --- a/docs/html/node8.html +++ b/docs/html/node8.html @@ -1,6 +1,6 @@ - + Optional third party libraries @@ -9,7 +9,7 @@ - + @@ -92,7 +92,7 @@ for multilevel preconditioners may change to reflect their presence.
      SuperLU_Dist
      [18] + HREF="node36.html#SUPERLUDIST">19] 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. diff --git a/docs/html/node9.html b/docs/html/node9.html index df5e4663..212118d8 100644 --- a/docs/html/node9.html +++ b/docs/html/node9.html @@ -1,6 +1,6 @@ - + Configuration options @@ -9,7 +9,7 @@ - + diff --git a/docs/html/userhtml.html b/docs/html/userhtml.html index f026ce9d..1f0d2ee7 100644 --- a/docs/html/userhtml.html +++ b/docs/html/userhtml.html @@ -1,6 +1,6 @@ - + userhtml @@ -9,7 +9,7 @@ - + @@ -69,17 +69,15 @@ based on PSBLAS


      -Software version: 2.1 +Software version: 2.2
      -July 31, 2017 +July 31, 2018

      - - - +


      diff --git a/docs/mld2p4-2.1-guide.pdf b/docs/mld2p4-2.2-guide.pdf similarity index 94% rename from docs/mld2p4-2.1-guide.pdf rename to docs/mld2p4-2.2-guide.pdf index 859ae4c5..664a13a8 100644 --- a/docs/mld2p4-2.1-guide.pdf +++ b/docs/mld2p4-2.2-guide.pdf @@ -24,7 +24,7 @@ BT /F17 11.9552 Tf 218.644 -79.389 Td [(P)31(asqua)-375(D'Am)31(bra)]TJ/F37 11.9552 Tf -22.655 -13.947 Td [(IA)27(C-CNR,)-326(Naples,)-326(Italy)]TJ/F17 11.9552 Tf 11.494 -29.39 Td [(Daniela)-375(di)-375(Sera\014no)]TJ/F37 11.9552 Tf -181.63 -13.948 Td [(Univ)27(ersit)27(y)-326(of)-326(Campania)-326(\134Luigi)-327(V)82(an)27(vitelli",)-326(Caserta,)-326(Italy)]TJ/F17 11.9552 Tf 179.561 -29.39 Td [(Salv)62(atore)-375(Filipp)-31(one)]TJ/F37 11.9552 Tf -134.787 -13.947 Td [(Cran\014eld)-326(Univ)27(ersit)27(y)82(,)-326(Cran\014eld,)-327(United)-326(Kingdom)]TJ 0 g 0 G 0 g 0 G - 141.76 -78.924 Td [(Soft)27(w)28(are)-327(v)27(ersion)1(:)-436(2.1)]TJ 38.924 -13.948 Td [(July)-326(31,)-327(2017)]TJ + 141.76 -78.924 Td [(Soft)27(w)28(are)-327(v)27(ersion)1(:)-436(2.2)]TJ 38.924 -13.948 Td [(July)-326(31,)-327(2018)]TJ 0 g 0 G 0 g 0 G ET @@ -722,11 +722,11 @@ BT 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-269(23)]TJ + [-269(24)]TJ 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-268(21)]TJ + [-268(22)]TJ 0 g 0 G [(]\051,)-281(to)-269(b)-27(e)-269(used)-268(in)-268(the)-269(iterativ)28(e)]TJ 0 -13.549 Td [(solution)-333(of)-334(lin)1(e)-1(ar)-333(systems,)]TJ/F22 10.9091 Tf 186.98 -16.123 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 @@ -738,7 +738,7 @@ BT 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-263(19)]TJ + [-263(20)]TJ 0 g 0 G [(])]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)-394(the)-394(m)28(ultilev)28(el)-394(h)28(ybrid)-394(Sc)28(h)27(w)28(arz)-394(ones;)-424(a)-394(sp)-28(eci\014c)-394(cycle)-394(is)-394(implemen)28(ted)-394(to)-394(obtain)]TJ 0 -13.55 Td [(m)28(ultilev)28(el)-498(additiv)28(e)-498(Sc)28(h)28(w)28(arz)-498(preconditioners.)-937(The)-497(Jacobi,)-539(h)28(ybrid)-498(forw)28(ard/bac)28(kw)28(ard)]TJ 0 -13.549 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 @@ -746,7 +746,7 @@ BT 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-236(25)]TJ + [-236(26)]TJ 0 g 0 G [(])]TJ 0 -13.55 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 -13.549 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 -14.408 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 -13.55 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 -14.407 Td [(MLD2P4)-229(has)-230(b)-27(e)-1(en)-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 @@ -908,7 +908,7 @@ BT 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-424(17)]TJ + [-424(18)]TJ 0 g 0 G [(])-425(Man)28(y)-425(v)28(endors)-424(pro)27(v)1(ide)-425(optimized)-425(v)28(ersions)-425(of)-424(BLAS;)-425(if)-424(no)-425(v)28(endor)]TJ -11.105 -13.55 Td [(v)28(ersion)-523(is)-524(a)28(v)56(ailable)-523(for)-523(a)-524(giv)28(en)-523(platform,)-570(the)-524(A)84(TLAS)-523(soft)28(w)27(are)-523(\050)]TJ 0 1 0 0 k 0 1 0 0 K @@ -928,11 +928,11 @@ BT 0 g 0 G /F15 10.9091 Tf 30.697 0 Td [([)]TJ 1 0 0 rg 1 0 0 RG - [(16)]TJ + [(17)]TJ 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-340(22)]TJ + [-340(23)]TJ 0 g 0 G [(])-340(A)-341(v)28(ersion)-340(of)-340(MPI)-340(is)-341(a)28(v)56(ailable)-341(on)-340(most)-340(high-p)-28(erformance)-340(computing)-340(sys-)]TJ -3.424 -13.549 Td [(tems.)]TJ 0 g 0 G @@ -1018,7 +1018,7 @@ BT 0 g 0 G /F15 10.9091 Tf 28.388 0 Td [([)]TJ 1 0 0 rg 1 0 0 RG - [(18)]TJ + [(19)]TJ 0 g 0 G [(])-417(A)-416(sparse)-417(LU)-416(factorization)-417(pac)28(k)55(age)-416(a)28(v)55(ailable)-416(from)-417(the)-417(same)-416(site)]TJ -55.002 -13.549 Td [(as)-366(Sup)-28(erLU;)-366(it)-367(pro)28(vides)-366(parallel)-366(factorization)-367(and)-366(triangular)-366(system)-367(solution)-366(for)]TJ 0 -13.549 Td [(double)-384(precision)-384(real)-384(and)-384(complex)-384(data.)-596(W)83(e)-384(tested)-384(v)28(ersions)-384(3.3)-384(and)-384(4.2.)-596(If)-384(y)28(ou)]TJ 0 -13.549 Td [(installed)-384(BLAS)-384(f)1(rom)-384(A)83(TLAS,)-384(remem)28(b)-28(er)-384(to)-383(de\014ne)-384(the)-384(BLASLIB)-384(v)56(ariable)-384(in)-384(the)]TJ 0 -13.549 Td [(mak)28(e.inc)-401(\014le)-401(and)-401(to)-401(add)-402(t)1(he)]TJ/F45 10.9091 Tf 142.011 0 Td [(-std=c99)]TJ/F15 10.9091 Tf 50.193 0 Td [(option)-401(to)-401(the)-401(C)-401(compiler)-401(options.)-648(Note)]TJ -192.204 -13.55 Td [(that)-432(this)-432(library)-432(requires)-433(t)1(he)-433(P)28(arMETIS)-432(library)-432(for)-432(parallel)-432(graph)-432(partitioning)]TJ 0 -13.549 Td [(and)-374(\014ll-reducing)-374(matrix)-374(ordering,)-385(a)28(v)56(ailable)-374(from)]TJ 0 1 0 0 k 0 1 0 0 K @@ -1891,11 +1891,11 @@ BT 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-313(23)]TJ + [-313(24)]TJ 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-313(21)]TJ + [-313(22)]TJ 0 g 0 G [(])]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 @@ -1903,11 +1903,11 @@ BT 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-298(25)]TJ + [-298(26)]TJ 0 g 0 G [(],)-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 - [(24)]TJ + [(25)]TJ 0 g 0 G [(].)-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 @@ -1955,7 +1955,7 @@ BT 0 g 0 G -398.511 -19.516 Td [(where)]TJ/F22 10.9091 Tf 32.496 0 Td [(A)]TJ/F15 10.9091 Tf 12.796 0 Td [(=)-423(\050)]TJ/F22 10.9091 Tf 17.342 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 8.857 0 Td [(2)]TJ/F34 10.9091 Tf 11.888 0 Td [(R)]TJ/F23 7.9701 Tf 7.878 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 10.223 -3.959 Td [(is)-421(a)-420(nonsingular)-420(sparse)-421(matrix;)-464(for)-420(ease)-421(of)-420(presen)28(tation)-421(w)28(e)]TJ -126.237 -13.549 Td [(assume)]TJ/F22 10.9091 Tf 37.697 0 Td [(A)]TJ/F15 10.9091 Tf 11.818 0 Td [(has)-333(a)-334(symmetric)-333(sparsit)28(y)-334(pattern.)]TJ -32.579 -13.549 Td [(Let)-468(us)-469(consider)-468(as)-468(\014nest)-469(ind)1(e)-1(x)-468(space)-468(the)-468(s)-1(et)-468(of)-468(ro)28(w)-469(\050column\051)-468(indices)-468(of)]TJ/F22 10.9091 Tf 361.856 0 Td [(A)]TJ/F15 10.9091 Tf 8.182 0 Td [(,)-502(i.e.,)]TJ -386.974 -13.55 Td [(\012)-377(=)]TJ/F25 10.9091 Tf 24.593 0 Td [(f)]TJ/F15 10.9091 Tf 5.454 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 5.455 0 Td [(.)-623(An)28(y)-393(algebraic)-393(m)27(ultilev)28(el)-393(preconditioners)-393(implemen)28(ted)-393(in)-393(MLD2P4)]TJ -82.05 -13.549 Td [(generates)-333(a)-334(hierarc)28(h)28(y)-333(of)-334(index)-333(spaces)-333(and)-334(a)-333(corresp)-28(onding)-333(hierarc)28(h)28(y)-334(of)-333(matrices,)]TJ 81.377 -23.491 Td [(\012)]TJ/F20 7.9701 Tf 7.879 4.505 Td [(1)]TJ/F25 10.9091 Tf 7.762 -4.505 Td [(\021)]TJ/F15 10.9091 Tf 11.515 0 Td [(\012)]TJ/F25 10.9091 Tf 10.909 0 Td [(\033)]TJ/F15 10.9091 Tf 11.515 0 Td [(\012)]TJ/F20 7.9701 Tf 7.879 4.505 Td [(2)]TJ/F25 10.9091 Tf 7.763 -4.505 Td [(\033)]TJ/F22 10.9091 Tf 11.515 0 Td [(:)-167(:)-166(:)]TJ/F25 10.9091 Tf 15.757 0 Td [(\033)]TJ/F15 10.9091 Tf 11.516 0 Td [(\012)]TJ/F23 7.9701 Tf 7.878 4.505 Td [(nl)-12(ev)]TJ/F22 10.9091 Tf 16.597 -4.505 Td [(;)-1167(A)]TJ/F20 7.9701 Tf 23.939 4.505 Td [(1)]TJ/F25 10.9091 Tf 7.763 -4.505 Td [(\021)]TJ/F22 10.9091 Tf 11.515 0 Td [(A;)-167(A)]TJ/F20 7.9701 Tf 21.212 4.505 Td [(2)]TJ/F22 10.9091 Tf 4.732 -4.505 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(A)]TJ/F23 7.9701 Tf 32.424 4.505 Td [(nl)-12(ev)]TJ/F22 10.9091 Tf 16.597 -4.505 Td [(;)]TJ/F15 10.9091 Tf -328.044 -23.49 Td [(b)28(y)-282(using)-282(the)-282(information)-282(con)28(tained)-282(in)]TJ/F22 10.9091 Tf 180.373 0 Td [(A)]TJ/F15 10.9091 Tf 8.182 0 Td [(,)-292(without)-282(assuming)-282(an)27(y)-282(kno)28(wledge)-282(of)-282(the)-282(geom-)]TJ -188.555 -13.55 Td [(etry)-295(of)-294(the)-295(problem)-294(from)-295(whic)28(h)]TJ/F22 10.9091 Tf 150.888 0 Td [(A)]TJ/F15 10.9091 Tf 11.395 0 Td [(originates.)-431(A)-295(v)28(ector)-295(space)]TJ/F34 10.9091 Tf 126.348 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/F15 10.9091 Tf 8.285 -2.553 Td [(is)-295(asso)-27(ciate)-1(d)-294(with)-295(\012)]TJ/F23 7.9701 Tf 94.368 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(,)]TJ -409.421 -13.549 Td [(where)]TJ/F22 10.9091 Tf 31.714 0 Td [(n)]TJ/F23 7.9701 Tf 6.548 -1.777 Td [(k)]TJ/F15 10.9091 Tf 8.925 1.777 Td [(is)-349(the)-349(size)-348(of)-349(\012)]TJ/F23 7.9701 Tf 71.401 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(.)-491(F)84(or)-349(all)]TJ/F22 10.9091 Tf 43.449 0 Td [(k)-335(<)-304(nl)-19(ev)]TJ/F15 10.9091 Tf 41.907 0 Td [(,)-353(a)-348(restriction)-349(op)-28(erator)-349(and)-348(a)-349(prolongation)]TJ -209.064 -13.549 Td [(one)-333(are)-334(built,)-333(whic)28(h)-333(connect)-334(t)28(w)28(o)-334(l)1(e)-1(v)28(els)]TJ/F22 10.9091 Tf 188.273 0 Td [(k)]TJ/F15 10.9091 Tf 9.659 0 Td [(and)]TJ/F22 10.9091 Tf 21.212 0 Td [(k)]TJ/F15 10.9091 Tf 8.447 0 Td [(+)-222(1:)]TJ/F22 10.9091 Tf -101.285 -23.491 Td [(P)]TJ/F23 7.9701 Tf 8.519 4.504 Td [(k)]TJ/F25 10.9091 Tf 8.15 -4.504 Td [(2)]TJ/F34 10.9091 Tf 10.303 0 Td [(R)]TJ/F23 7.9701 Tf 7.879 4.504 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.405 Td [(k)]TJ/F26 7.9701 Tf 4.573 1.405 Td [(\002)]TJ/F23 7.9701 Tf 6.587 0 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.405 Td [(k)]TJ/F21 5.9776 Tf 4.075 0 Td [(+1)]TJ/F22 10.9091 Tf 10.239 -3.099 Td [(;)-1167(R)]TJ/F23 7.9701 Tf 24.125 4.504 Td [(k)]TJ/F25 10.9091 Tf 8.15 -4.504 Td [(2)]TJ/F34 10.9091 Tf 10.303 0 Td [(R)]TJ/F23 7.9701 Tf 7.879 4.504 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.405 Td [(k)]TJ/F21 5.9776 Tf 4.075 0 Td [(+1)]TJ/F26 7.9701 Tf 9.741 1.405 Td [(\002)]TJ/F23 7.9701 Tf 6.587 0 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.405 Td [(k)]TJ/F15 10.9091 Tf 5.071 -3.099 Td [(;)]TJ -283.114 -23.491 Td [(the)-282(matrix)]TJ/F22 10.9091 Tf 53.146 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.621 0 Td [(+1)]TJ/F15 10.9091 Tf 14.392 -3.959 Td [(is)-282(computed)-281(b)27(y)-281(using)-282(the)-282(previou)1(s)-282(op)-28(erators)-282(according)-281(to)-282(the)-282(Galerkin)]TJ -80.341 -13.549 Td [(approac)28(h,)-333(i.e.,)]TJ/F22 10.9091 Tf 165.163 -13.549 Td [(A)]TJ/F23 7.9701 Tf 8.182 4.504 Td [(k)]TJ/F20 7.9701 Tf 4.621 0 Td [(+1)]TJ/F15 10.9091 Tf 14.349 -4.504 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 4.504 Td [(k)]TJ/F22 10.9091 Tf 5.12 -4.504 Td [(A)]TJ/F23 7.9701 Tf 8.181 4.504 Td [(k)]TJ/F22 10.9091 Tf 5.12 -4.504 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 -244.258 -19.516 Td [(In)-476(the)-477(curren)28(t)-476(implemen)28(tation)-477(of)-476(MLD2P4)-476(w)28(e)-477(ha)28(v)28(e)]TJ/F22 10.9091 Tf 261.479 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 3.958 Td [(k)]TJ/F15 10.9091 Tf 10.749 -3.958 Td [(=)-516(\050)]TJ/F22 10.9091 Tf 18.358 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.958 Td [(k)]TJ/F15 10.9091 Tf 5.119 -3.958 Td [(\051)]TJ/F23 7.9701 Tf 4.243 3.958 Td [(T)]TJ/F15 10.9091 Tf 11.801 -3.958 Td [(A)-476(smo)-28(other)-476(with)]TJ -328.636 -13.55 Td [(iteration)-441(matrix)]TJ/F22 10.9091 Tf 82.101 0 Td [(M)]TJ/F23 7.9701 Tf 11.773 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.928 -3.959 Td [(is)-441(set)-441(up)-440(at)-441(eac)28(h)-441(lev)28(el)]TJ/F22 10.9091 Tf 113.517 0 Td [(k)-488(<)-457(nl)-20(ev)]TJ/F15 10.9091 Tf 45.251 0 Td [(,)-468(and)-440(a)-441(solv)28(er)-441(is)-441(set)-441(up)-440(at)-441(the)]TJ -262.57 -13.549 Td [(coarsest)-369(lev)28(el,)-378(so)-368(that)-369(they)-369(are)-369(ready)-368(for)-369(application)-369(\050for)-368(example,)-378(setting)-369(up)-368(a)-369(solv)28(er)]TJ 0 -13.549 Td [(based)-292(on)-291(the)]TJ/F22 10.9091 Tf 62.938 0 Td [(LU)]TJ/F15 10.9091 Tf 19.243 0 Td [(factorization)-292(means)-291(computing)-292(and)-291(storing)-292(the)]TJ/F22 10.9091 Tf 224.422 0 Td [(L)]TJ/F15 10.9091 Tf 10.605 0 Td [(and)]TJ/F22 10.9091 Tf 20.757 0 Td [(U)]TJ/F15 10.9091 Tf 11.819 0 Td [(factors\051.)-431(The)]TJ -349.784 -13.549 Td [(construction)-345(of)-344(the)-345(hierarc)28(h)27(y)-344(of)-345(AMG)-345(comp)-28(on)1(e)-1(n)28(ts)-345(d)1(e)-1(scrib)-27(ed)-345(so)-345(far)-345(corresp)-27(onds)-345(to)-345(the)]TJ 0 -13.549 Td [(so-called)-333(build)-334(ph)1(as)-1(e)-333(of)-333(the)-334(p)1(rec)-1(on)1(ditioner.)]TJ 16.936 -13.55 Td [(The)-359(com)-1(p)-27(onen)28(ts)-360(pro)-28(du)1(c)-1(ed)-359(in)-359(the)-360(build)-359(phase)-359(ma)27(y)-359(b)-28(e)-359(com)28(bined)-360(in)-359(sev)28(eral)-360(w)28(a)28(ys)-360(to)]TJ -16.936 -13.549 Td [(obtain)-306(di\013eren)28(t)-306(m)27(ultilev)28(el)-306(preconditioners;)-315(this)-306(is)-306(done)-307(in)-306(the)-306(application)-306(phase,)-312(i)1(.e)-1(.)1(,)-312(in)]TJ 0 -13.549 Td [(the)-391(computation)-391(of)-391(a)-391(v)27(ector)-391(of)-391(t)28(yp)-28(e)]TJ/F22 10.9091 Tf 178.081 0 Td [(w)]TJ/F15 10.9091 Tf 12.185 0 Td [(=)]TJ/F22 10.9091 Tf 12.566 0 Td [(B)]TJ/F26 7.9701 Tf 8.822 3.959 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 4.732 -3.959 Td [(v)]TJ/F15 10.9091 Tf 5.68 0 Td [(,)-406(where)]TJ/F22 10.9091 Tf 39.631 0 Td [(B)]TJ/F15 10.9091 Tf 13.089 0 Td [(denotes)-391(the)-391(preconditioner,)]TJ -281.372 -13.549 Td [(usually)-334(with)1(in)-334(an)-334(iteration)-333(of)-334(a)-334(Kr)1(ylo)27(v)-333(solv)27(er)-333([)]TJ 1 0 0 rg 1 0 0 RG - [(20)]TJ + [(21)]TJ 0 g 0 G [(].)-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 @@ -1967,7 +1967,7 @@ BT 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 - [(19)]TJ + [(20)]TJ 0 g 0 G [(].)]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)-374(the)-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 @@ -1975,7 +1975,7 @@ BT 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-319(25)]TJ + [-319(26)]TJ 0 g 0 G [(].)-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 @@ -2038,7 +2038,7 @@ BT 0 g 0 G [-500(application)-333(of)]TJ/F22 10.9091 Tf 82.727 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.958 Td [(k)]TJ/F15 10.9091 Tf 8.756 -3.958 Td [(and)]TJ/F22 10.9091 Tf 21.212 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 3.958 Td [(k)]TJ/F15 10.9091 Tf 8.15 -3.958 Td [(=)-278(\050)]TJ/F22 10.9091 Tf 15.757 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.958 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.958 Td [(\051)]TJ/F23 7.9701 Tf 4.242 3.958 Td [(T)]TJ/F15 10.9091 Tf 10.241 -3.958 Td [(to)-333(build)]TJ/F22 10.9091 Tf 41.212 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F15 10.9091 Tf 11.319 -3.958 Td [(.)]TJ -243.343 -23.467 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)-478(in)-478([)]TJ 1 0 0 rg 1 0 0 RG - [(25)]TJ + [(26)]TJ 0 g 0 G [(])-479(is)-478(used.)-879(In)-479(this)-478(algorithm,)-514(eac)27(h)-478(index)]TJ/F22 10.9091 Tf 271.801 0 Td [(j)]TJ/F25 10.9091 Tf 10.783 0 Td [(2)]TJ/F15 10.9091 Tf 12.939 0 Td [(\012)]TJ/F23 7.9701 Tf 7.879 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F15 10.9091 Tf 16.537 -3.959 Td [(corresp)-28(onds)-478(to)-478(an)]TJ -324.561 -13.549 Td [(aggregate)-374(\012)]TJ/F23 7.9701 Tf 57.445 3.958 Td [(k)]TJ 0 -7.014 Td [(j)]TJ/F15 10.9091 Tf 9.201 3.056 Td [(of)-374(\012)]TJ/F23 7.9701 Tf 20.747 3.958 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.958 Td [(,)-384(consisting)-374(of)-374(a)-374(s)-1(u)1(itably)-374(c)27(hosen)-374(index)]TJ/F22 10.9091 Tf 187.709 0 Td [(i)]TJ/F25 10.9091 Tf 7.529 0 Td [(2)]TJ/F15 10.9091 Tf 11.044 0 Td [(\012)]TJ/F23 7.9701 Tf 7.879 3.958 Td [(k)]TJ/F15 10.9091 Tf 9.201 -3.958 Td [(and)-374(indices)-374(that)-374(are)]TJ -315.875 -13.55 Td [(\050usually\051)-333(con)28(tained)-334(in)-333(a)-333(strongly-coupled)-334(n)1(e)-1(igh)28(b)-27(oro)-28(o)-28(d)-333(of)]TJ/F22 10.9091 Tf 274.122 0 Td [(i)]TJ/F15 10.9091 Tf 3.758 0 Td [(,)-333(i.e.,)]TJ -194.032 -31.214 Td [(\012)]TJ/F23 7.9701 Tf 7.879 4.505 Td [(k)]TJ 0 -7.202 Td [(j)]TJ/F25 10.9091 Tf 8.15 2.697 Td [(\032)-278(N)]TJ/F23 7.9701 Tf 22.073 4.505 Td [(k)]TJ -1.607 -7.202 Td [(i)]TJ/F15 10.9091 Tf 6.727 2.697 Td [(\050)]TJ/F22 10.9091 Tf 4.242 0 Td [(\022)]TJ/F15 10.9091 Tf 5.425 0 Td [(\051)-278(=)]TJ/F28 10.9091 Tf 18.788 15.382 Td [(\032)]TJ/F22 10.9091 Tf 8.181 -15.382 Td [(r)]TJ/F25 10.9091 Tf 8.255 0 Td [(2)]TJ/F15 10.9091 Tf 10.303 0 Td [(\012)]TJ/F23 7.9701 Tf 7.879 4.505 Td [(k)]TJ/F15 10.9091 Tf 8.15 -4.505 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 4.505 Td [(k)]TJ 0 -7.202 Td [(ir)]TJ/F25 10.9091 Tf 7.438 2.697 Td [(j)]TJ/F22 10.9091 Tf 6.061 0 Td [(>)-278(\022)]TJ/F28 10.9091 Tf 16.939 13.424 Td [(q)]TJ ET @@ -2053,7 +2053,7 @@ BT 0 g 0 G -398.512 -31.214 Td [(for)-431(a)-430(giv)27(en)-430(threshold)]TJ/F22 10.9091 Tf 106.493 0 Td [(\022)]TJ/F25 10.9091 Tf 10.225 0 Td [(2)]TJ/F15 10.9091 Tf 12.074 0 Td [([0)]TJ/F22 10.9091 Tf 8.485 0 Td [(;)]TJ/F15 10.9091 Tf 4.848 0 Td [(1])-431(\050see)-431([)]TJ 1 0 0 rg 1 0 0 RG - [(25)]TJ + [(26)]TJ 0 g 0 G [(])-430(for)-431(the)-431(details\051.)-736(Since)-431(this)-431(algorithm)-430(has)-431(a)]TJ -142.125 -13.549 Td [(sequen)28(tial)-382(n)1(ature,)-394(a)-381(decoupled)-381(v)28(e)-1(r)1(s)-1(i)1(on)-382(of)-381(it)-381(is)-382(appli)1(e)-1(d,)-393(where)-381(eac)28(h)-382(pro)-27(ces)-1(sor)-381(indep)-28(en-)]TJ 0 -13.549 Td [(den)28(tly)-433(exe)-1(cutes)-433(the)-433(algorithm)-433(on)-434(the)-433(set)-433(of)-434(indices)-433(assigned)-433(to)-434(it)-433(in)-433(the)-433(initial)-434(data)]TJ 0 -13.549 Td [(distribution.)-442(Thi)1(s)-326(v)28(ersion)-325(is)-326(em)28(barrassingly)-325(parallel,)-327(since)-325(it)-326(do)-27(es)-326(not)-325(require)-325(an)27(y)-325(data)]TJ 0 -13.55 Td [(comm)28(unication.)-522(On)-359(the)-359(other)-359(hand,)-365(it)-360(ma)28(y)-359(pro)-28(du)1(c)-1(e)-359(some)-359(non)28(uniform)-359(aggregates)-359(and)]TJ 0 -13.549 Td [(is)-295(strongly)-295(dep)-27(enden)27(t)-294(on)-295(the)-295(n)28(um)28(b)-28(er)-295(of)-295(pro)-27(cess)-1(or)1(s)-295(and)-295(on)-295(the)-295(initial)-294(partitioning)-295(of)-295(the)]TJ 0 -13.549 Td [(matrix)]TJ/F22 10.9091 Tf 35.775 0 Td [(A)]TJ/F15 10.9091 Tf 8.182 0 Td [(.)-524(Nev)28(e)-1(rt)1(hele)-1(ss,)-366(this)-360(parallel)-360(algorithm)-360(has)-360(b)-27(ee)-1(n)-359(c)27(hosen)-360(for)-359(MLD2P4,)-367(since)-360(it)]TJ -43.957 -13.549 Td [(has)-333(b)-28(een)-333(s)-1(h)1(o)27(wn)-333(to)-333(pro)-28(duce)-333(go)-28(o)-28(d)-333(results)-334(in)-333(practice)-333([)]TJ 1 0 0 rg 1 0 0 RG @@ -2065,7 +2065,7 @@ BT 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-334(24)]TJ + [-334(25)]TJ 0 g 0 G [(].)]TJ 0 g 0 G @@ -2093,7 +2093,7 @@ BT 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-293(23)]TJ + [-293(24)]TJ 0 g 0 G [(].)-431(A)-294(simple)-293(c)28(hoice)]TJ 0 -13.549 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 118.068 -22.041 Td [(S)]TJ/F23 7.9701 Tf 7.318 4.505 Td [(k)]TJ/F15 10.9091 Tf 8.15 -4.505 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 7.182 4.505 Td [(k)]TJ/F15 10.9091 Tf 5.119 -4.505 Td [(\050)]TJ/F22 10.9091 Tf 4.243 0 Td [(D)]TJ/F23 7.9701 Tf 9.335 4.505 Td [(k)]TJ/F15 10.9091 Tf 5.119 -4.505 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 4.733 -4.505 Td [(A)]TJ/F23 7.9701 Tf 8.181 4.505 Td [(k)]TJ 0 -7.202 Td [(F)]TJ/F22 10.9091 Tf 7.034 2.697 Td [(;)]TJ/F15 10.9091 Tf -258.582 -22.04 Td [(where)]TJ/F22 10.9091 Tf 32.045 0 Td [(D)]TJ/F23 7.9701 Tf 9.335 3.958 Td [(k)]TJ/F15 10.9091 Tf 9.255 -3.958 Td [(is)-379(the)-379(diagonal)-379(matrix)-379(with)-379(the)-379(s)-1(ame)-379(diagonal)-379(en)28(tries)-379(as)]TJ/F22 10.9091 Tf 276.813 0 Td [(A)]TJ/F23 7.9701 Tf 8.181 3.958 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.958 Td [(,)]TJ/F22 10.9091 Tf 7.291 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.958 Td [(k)]TJ 0 -7.191 Td [(F)]TJ/F15 10.9091 Tf 10.896 3.233 Td [(=)-354(\050)-14(\026)]TJ/F22 10.9091 Tf 16.59 0 Td [(a)]TJ/F23 7.9701 Tf 5.766 3.958 Td [(k)]TJ 0 -7.014 Td [(ij)]TJ/F15 10.9091 Tf 7.266 3.056 Td [(\051)-379(is)]TJ -396.74 -13.55 Td [(the)-333(\014ltered)-334(matrix)-333(de\014ned)-333(as)]TJ 48.968 -26.039 Td [(\026)]TJ/F22 10.9091 Tf -0.156 0 Td [(a)]TJ/F23 7.9701 Tf 5.766 4.505 Td [(k)]TJ 0 -7.202 Td [(ij)]TJ/F15 10.9091 Tf 10.296 2.697 Td [(=)]TJ/F28 10.9091 Tf 11.515 15.382 Td [(\032)]TJ/F22 10.9091 Tf 13.163 -8.324 Td [(a)]TJ/F23 7.9701 Tf 5.766 3.959 Td [(k)]TJ 0 -7.015 Td [(ij)]TJ/F15 10.9091 Tf 17.228 3.056 Td [(if)]TJ/F22 10.9091 Tf 10 0 Td [(j)]TJ/F25 10.9091 Tf 8.147 0 Td [(2)-278(N)]TJ/F23 7.9701 Tf 20.862 3.959 Td [(k)]TJ -1.608 -7.015 Td [(i)]TJ/F15 10.9091 Tf 6.728 3.056 Td [(\050)]TJ/F22 10.9091 Tf 4.242 0 Td [(\022)]TJ/F15 10.9091 Tf 5.424 0 Td [(\051)]TJ/F22 10.9091 Tf 4.243 0 Td [(;)]TJ/F15 10.9091 Tf -81.032 -14.09 Td [(0)-1608(otherwise)]TJ/F22 10.9091 Tf 67.934 0 Td [(;)]TJ/F15 10.9091 Tf 27.153 7.032 Td [(\050)]TJ/F22 10.9091 Tf 4.243 0 Td [(j)]TJ/F25 10.9091 Tf 8.147 0 Td [(6)]TJ/F15 10.9091 Tf 0 0 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(i)]TJ/F15 10.9091 Tf 3.758 0 Td [(\051)]TJ/F22 10.9091 Tf 4.243 0 Td [(;)]TJ/F15 10.9091 Tf 26.822 0 Td [(\026)]TJ/F22 10.9091 Tf -0.156 0 Td [(a)]TJ/F23 7.9701 Tf 5.767 4.505 Td [(k)]TJ 0 -7.202 Td [(ii)]TJ/F15 10.9091 Tf 9.294 2.697 Td [(=)]TJ/F22 10.9091 Tf 11.516 0 Td [(a)]TJ/F23 7.9701 Tf 5.766 4.505 Td [(k)]TJ 0 -7.202 Td [(ii)]TJ/F25 10.9091 Tf 8.689 2.697 Td [(\000)]TJ/F28 10.9091 Tf 10.909 10.364 Td [(X)]TJ/F23 7.9701 Tf 1.202 -23.717 Td [(j)]TJ/F26 7.9701 Tf 3.884 0 Td [(6)]TJ/F20 7.9701 Tf 0 0 Td [(=)]TJ/F23 7.9701 Tf 6.586 0 Td [(i)]TJ/F15 10.9091 Tf 4.085 13.353 Td [(\050)]TJ/F22 10.9091 Tf 4.243 0 Td [(a)]TJ/F23 7.9701 Tf 5.766 4.505 Td [(k)]TJ 0 -7.202 Td [(ij)]TJ/F25 10.9091 Tf 9.69 2.697 Td [(\000)]TJ/F15 10.9091 Tf 11.065 0 Td [(\026)]TJ/F22 10.9091 Tf -0.156 0 Td [(a)]TJ/F23 7.9701 Tf 5.766 4.505 Td [(k)]TJ 0 -7.202 Td [(ij)]TJ/F15 10.9091 Tf 7.265 2.697 Td [(\051)]TJ/F22 10.9091 Tf 4.243 0 Td [(;)]TJ 0 g 0 G @@ -2105,11 +2105,11 @@ BT 0 g 0 G [(].)]TJ -397.906 -13.549 Td [(In)-288(MLD2P4)-288(this)-287(appro)27(ximation)-287(is)-288(obtained)-288(b)28(y)-288(using)]TJ/F25 10.9091 Tf 250.336 0 Td [(k)]TJ/F22 10.9091 Tf 5.455 0 Td [(A)]TJ/F23 7.9701 Tf 8.181 3.959 Td [(k)]TJ 0 -7.192 Td [(F)]TJ/F25 10.9091 Tf 7.034 3.233 Td [(k)]TJ/F26 7.9701 Tf 5.454 -1.636 Td [(1)]TJ/F15 10.9091 Tf 12.107 1.636 Td [(as)-288(an)-288(estimate)-288(of)]TJ/F22 10.9091 Tf 82.683 0 Td [(\032)]TJ/F23 7.9701 Tf 5.641 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.119 -3.959 Td [(.)-429(Note)]TJ -382.01 -13.549 Td [(that)-373(for)-373(systems)-374(coming)-373(from)-373(uniformly)-373(elliptic)-373(problems,)-384(\014)1(ltering)-374(the)-373(matrix)]TJ/F22 10.9091 Tf 379.26 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.191 -3.959 Td [(has)]TJ -396.633 -13.549 Td [(little)-306(or)-306(no)-306(e\013ect,)-312(and)]TJ/F22 10.9091 Tf 106.178 0 Td [(A)]TJ/F23 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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{GrHeJi:16} +S. Gratton, P. Henon, P. Jiranek and X. Vasseur, + {\em Reducing complexity of algebraic multigrid by aggregation}, +Numerical Lin. Algebra with Applications, 2016, 23:501-518 + % \bibitem{MPI2} W.~Gropp, S.~Huss-Lederman, A.~Lumsdaine, E.~Lusk, B.~Nitzberg, W.~Saphir, M.~Snir, diff --git a/docs/src/userguide.tex b/docs/src/userguide.tex index 7584dd9e..70e4fab1 100644 --- a/docs/src/userguide.tex +++ b/docs/src/userguide.tex @@ -27,7 +27,7 @@ \pdfcompresslevel=0 %-- 0 = none, 9 = best \pdfinfo{ %-- Info dictionary of PDF output /Author (PD, DdS, SF) /Title (MultiLevel Domain Decomposition Parallel Preconditioners Package - based on PSBLAS, V. 2.1) + based on PSBLAS, V. 2.2) /Subject (MultiLevel Domain Decomposition Parallel Preconditioners Package) /Keywords (Parallel Numerical Software, Algebraic Multilevel Preconditioners, Sparse Iterative Solvers, PSBLAS, MPI) /Creator (pdfLaTeX) @@ -125,9 +125,9 @@ based on PSBLAS} \vspace{\stretch{1}} \noindent\hspace*{\centeroffset}\makebox[0pt][l]{\begin{minipage}{\textwidth} \flushright -\large Software version: 2.1\\ +\large Software version: 2.2\\ %\today -\large July 31, 2017 +\large July 31, 2018 \end{minipage}} %\addtolength{\textwidth}{\centeroffset} \vspace{\stretch{2}} diff --git a/docs/src/userhtml.tex b/docs/src/userhtml.tex index 5a979980..86d4cf93 100644 --- a/docs/src/userhtml.tex +++ b/docs/src/userhtml.tex @@ -26,7 +26,7 @@ \pdfcompresslevel=0 %-- 0 = none, 9 = best \pdfinfo{ %-- Info dictionary of PDF output /Author (PD, DdS, SF) /Title (MultiLevel Domain Decomposition Parallel Preconditioners Package - based on PSBLAS, V. 2.1) + based on PSBLAS, V. 2.2) /Subject (MultiLevel Domain Decomposition Parallel Preconditioners Package) /Keywords (Parallel Numerical Software, Algebraic Multilevel Preconditioners, Sparse Iterative Solvers, PSBLAS, MPI) /Creator (pdfLaTeX) @@ -102,9 +102,9 @@ based on PSBLAS}\\[3ex] \large Cranfield University, Cranfield, United Kingdom \\[10ex] %\today -Software version: 2.1\\ +Software version: 2.2\\ %\today - July 31, 2017 + July 31, 2018 \clearpage \ \\ \thispagestyle{empty} diff --git a/docs/src/userinterface.tex b/docs/src/userinterface.tex index 0e3de52d..82a13bfa 100644 --- a/docs/src/userinterface.tex +++ b/docs/src/userinterface.tex @@ -319,10 +319,14 @@ be applied. of $A+A^T$.\\ \hline %\verb|mld_aggr_type_| \par \verb|'AGGR_TYPE'| & \verb|character(len=*)| \hspace*{-3mm} - & \textbf{\texttt{'VMB'}} & \textbf{\texttt{'VMB'}} - & Type of aggregation algorithm: currently, the scalar aggregation - algorithm by Van\v{e}k, Mandel and Brezina is implemented - \cite{VANEK_MANDEL_BREZINA}. \\ \hline + & \textbf{\texttt{'SOC1'}} & + \textbf{\texttt{'SOC1'}}, + \textbf{\texttt{'SOC2'}} + & Type of aggregation algorithm: currently, + we implement to measures of strength of + connection, the one by Van\v{e}k, Mandel + and Brezina~\cite{VANEK_MANDEL_BREZINA}, + and the one by Gratton et al~\cite{GrHeJi:16}. \\ \hline %\verb|mld_aggr_prol_| \par \verb|'AGGR_PROL'| & \verb|character(len=*)| \hspace*{-3mm} & \texttt{'SMOOTHED'}, \texttt{'UNSMOOTHED'} & \texttt{'SMOOTHED'}