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diff --git a/docs/html/img91.png b/docs/html/img91.png index 4a631dd6..18a953e2 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 ed89fe01..b58a4533 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 925d3460..cc5d0c13 100644 Binary files a/docs/html/img93.png and b/docs/html/img93.png differ diff --git a/docs/html/node13.html b/docs/html/node13.html index 7cfb275b..26b49b23 100644 --- a/docs/html/node13.html +++ b/docs/html/node13.html @@ -80,25 +80,25 @@ where $A=(a_{ij}) \in \mathbb{R}^{n \times n}$ is a nonsingular sparse matrix; for ease of presentation we assume $A$ has a symmetric sparsity pattern.

Let us consider 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 @@ -122,39 +122,39 @@ a hierarchy of index spaces and a corresponding hierarchy of matrices,

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 $k+1$: @@ -168,7 +168,7 @@ 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 @@ -178,7 +178,7 @@ R^k \in \mathbb{R}^{n_{k+1}\times n_k}; <BR CLEAR=

the matrix $A^{k+1}$ is computed by using the previous operators according to the Galerkin approach, i.e., @@ -192,7 +192,7 @@ A^{k+1}=R^kA^kP^k. --> \begin{displaymath} A^{k+1}=R^kA^kP^k. @@ -205,22 +205,22 @@ In the current implementation of MLD2P4 we have <SPAN CLASS=$R^k=(P^k)^T$ 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 (for example, setting up a solver based on the $LU$ factorization means computing and storing the $L$ and $U$ factors). The construction of the hierarchy of AMG components described so far corresponds to the so-called build phase of the preconditioner. @@ -262,8 +262,8 @@ end \begin{tabbing} \quad \=\quad \=\quad... ...[1mm] -\>endif [1mm] -\>return $u^k$ [1mm] +\>endif \\ [1mm] +\>return $u^k$\ \\ [1mm] end \end{tabbing}\end{minipage}}"> @@ -276,7 +276,7 @@ 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

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 [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. @@ -116,13 +116,13 @@ 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 [26] for the details). @@ -175,7 +175,7 @@ distribution. This version is embarrassingly parallel, since it does not require communication. On the other hand, it may produce some nonuniform aggregates and is strongly dependent on the number of processors and on the initial partitioning of the matrix $A$. Nevertheless, this parallel algorithm has been chosen for MLD2P4, since it has been shown to produce good results in practice @@ -186,7 +186,7 @@ MLD2P4, since it has been shown to produce good results in practice

    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, @@ -277,7 +277,7 @@ method [<A HREF=2,24]. A simple choice for $S^k$ is the damped Jacobi smoother: @@ -290,7 +290,7 @@ S^k = I - \omega^k (D^k)^{-1} A^k_F , --> \begin{displaymath} S^k = I - \omega^k (D^k)^{-1} A^k_F , @@ -299,17 +299,17 @@ S^k = I - \omega^k (D^k)^{-1} A^k_F , <BR CLEAR=

    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 @@ -344,7 +344,7 @@ a_{ij}^k & \m...

    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 db064fff..f45480f8 100644 --- a/docs/html/node15.html +++ b/docs/html/node15.html @@ -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 $\mathbb{R}^{n_{k,i}}$, i.e. $\lfloor 40 \sqrt[3]{n} \rfloor$, where $n$ is the dimension of the matrix at the finest level @@ -376,7 +376,7 @@ Currently, only the SYMDEC option applies decoupled aggregation to the sparsity pattern of $A+A^T$. @@ -449,12 +449,12 @@ Parameters defining the aggregation algorithm (continued). Any real

    number $\in [0, 1]$ 0.01 The threshold $\theta$ in the aggregation algorithm, see (3) in Section 4.2. @@ -643,7 +643,7 @@ number $\ge 0$ 0 Drop tolerance $t$ in the ILU( 0 Drop tolerance $t$ in the ILU( $y = op(B^{-1})\, x$, where 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 $B^{-1})$. diff --git a/docs/html/node3.html b/docs/html/node3.html index 02b8228b..b7f7bd1d 100644 --- a/docs/html/node3.html +++ b/docs/html/node3.html @@ -82,7 +82,7 @@ Ax=b,

    where $A$ is a square, real or complex, sparse matrix. The name of the package comes from its original implementation, containing diff --git a/docs/mld2p4-2.2-guide.pdf b/docs/mld2p4-2.2-guide.pdf index 664a13a8..29db2a71 100644 --- a/docs/mld2p4-2.2-guide.pdf +++ b/docs/mld2p4-2.2-guide.pdf @@ -10166,8 +10166,8 @@ endobj 737 0 obj << /Title (MultiLevel Domain Decomposition Parallel Preconditioners Package 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) /Producer ($Id: userguide.tex 2008-04-08 Pasqua D'Ambra, Daniela di Serafino, Salvatore Filippone$) /Author()/Title()/Subject()/Creator(LaTeX with hyperref package)/Producer(pdfTeX-1.40.17)/Keywords() -/CreationDate (D:20180514140750+01'00') -/ModDate (D:20180514140750+01'00') +/CreationDate (D:20180514143546+01'00') +/ModDate (D:20180514143546+01'00') /Trapped /False /PTEX.Fullbanner (This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016) kpathsea version 6.2.2) >> @@ -10580,7 +10580,7 @@ endobj /W [1 3 1] /Root 736 0 R /Info 737 0 R -/ID [<3899C85B41B440C76C6602BE660690F2> <3899C85B41B440C76C6602BE660690F2>] +/ID [<8C81DAEA73A339289C5AFD04D6D5A2A7> <8C81DAEA73A339289C5AFD04D6D5A2A7>] /Length 3695 >> stream