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@@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<TITLE>Smoothed Aggregation</TITLE>
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<H1><A NAME="SECTION00070000000000000000"></A><A NAME="sec:started"></A>
<H2><A NAME="SECTION00062000000000000000"></A><A NAME="sec:aggregation"></A>
<BR>
Getting Started
</H1><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">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 [<A
HREF="node27.html#PSBLASGUIDE">13</A>].
The following steps are required:
</FONT></FONT></FONT>
Smoothed Aggregation
</H2><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍␍In order to define the prolongator <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$">, used to compute␍the coarse-level matrix <IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img15.png"
ALT="$A^{k+1}$">, MLD2P4 uses the smoothed aggregation␍algorithm described in [<A
HREF="node29.html#BREZINA_VANEK">2</A>,<A
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>].␍The basic idea of this algorithm is to build a coarse set of indices␍<IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img26.png"
ALT="$\Omega^{k+1}$"> by suitably grouping the indices of <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$"> into disjoint␍subsets (aggregates), and to define the coarse-to-fine space transfer operator␍<IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"> 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:␍</FONT></FONT></FONT>
<OL>
<LI><I>Declare the preconditioner data structure</I>. It is a derived data type,
<code>mld_</code><I>x</I><code>prec_</code> <code>type</code>, where <I>x</I> may be <code>s</code>, <code>d</code>, <code>c</code>
or <code>z</code>, according to the basic data type of the sparse matrix
(<code>s</code> = real single precision; <code>d</code> = real double precision;
<code>c</code> = complex single precision; <code>z</code> = complex double precision).
This data structure is accessed by the user only through the MLD2P4 routines,
following an object-oriented approach.
<LI>aggregation of the indices of <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$"> to obtain <IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img26.png"
ALT="$\Omega^{k+1}$">;
</LI>
<LI><I>Allocate and initialize the preconditioner data structure, according to
a preconditioner type chosen by the user</I>. This is performed by the routine
<code>init</code>, 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&nbsp;<A HREF="#tab:precinit">1</A>, where the strings used by
<code>init</code> to identify the preconditioner types are also given.
Note that these strings are valid also if uppercase letters are substituted by
corresponding lowercase ones.
<LI>construction of the prolongator <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$">;
</LI>
<LI><I>Modify the selected preconditioner type, by properly setting
preconditioner parameters.</I> This is performed by the routine <code>set</code>.
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 <code>set</code> are given in
Section&nbsp;<A HREF="node14.html#sec:examples">5.1</A>; a complete list of all the
preconditioner parameters and their allowed and default values is provided in
Section&nbsp;<A HREF="node15.html#sec:userinterface">6</A>, Tables&nbsp;<A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>.
<LI>application of <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"> and <IMG
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img17.png"
ALT="$R^k=(P^k)^T$"> to build <IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img15.png"
ALT="$A^{k+1}$">.
</LI>
<LI><I>Build the preconditioner for a given matrix</I>. If the selected preconditioner
is multi-level, then two steps must be performed, as specified next.
<DL COMPACT>
<DT>4.1</DT>
<DD><I>Build the aggregation hierarchy for a given matrix.</I> This is
performed by the routine <code>hierarchy_build</code>.
</DD>
<DT>4.2</DT>
<DD><I>Build the preconditioner for a given matrix.</I> This is performed
by the routine <code>smoothers_build</code>.
</DD>
</DL>
If the selected preconditioner is one-level, it is built in a single step,
performed by the routine <code>bld</code>.
</LI>
<LI><I>Apply the preconditioner at each iteration of a Krylov solver.</I>
This is performed by the routine <code>aply</code>. When using the PSBLAS Krylov solvers,
this step is completely transparent to the user, since <code>aply</code> is called
by the PSBLAS routine implementing the Krylov solver (<code>psb_krylov</code>).
</LI>
<LI><I>Free the preconditioner data structure</I>. This is performed by
the routine <code>free</code>. This step is complementary to step 1 and should
be performed when the preconditioner is no more used.
</LI>
</OL><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">All the previous routines are available as methods of the preconditioner object.
A detailed description of them is given in Section&nbsp;<A HREF="node15.html#sec:userinterface">6</A>.
Examples showing the basic use of MLD2P4 are reported in Section&nbsp;<A HREF="node14.html#sec:examples">5.1</A>.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="532"></A>
<TABLE>
<CAPTION><STRONG>Table 1:</STRONG>
Preconditioner types, corresponding strings and default choices.
</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT"><SMALL>TYPE</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><SMALL>STRING</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232><SMALL>DEFAULT PRECONDITIONER</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT">No preconditioner</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'NOPREC'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Considered only to use the PSBLAS
Krylov solvers with no preconditioner.</TD>
</TR>
<TR><TD ALIGN="LEFT">Diagonal</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'DIAG'</code> or <code>'JACOBI'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Diagonal preconditioner.
For any zero diagonal entry of the matrix to be preconditioned,
the corresponding entry of the preconditioner is set to&nbsp;1.</TD>
</TR>
<TR><TD ALIGN="LEFT">Block Jacobi</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'BJAC'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Block-Jacobi with ILU(0) on the local blocks.</TD>
</TR>
<TR><TD ALIGN="LEFT">Additive Schwarz</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'AS'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Restricted Additive Schwarz (RAS),
with overlap&nbsp;1 and ILU(0) on the local blocks.</TD>
</TR>
<TR><TD ALIGN="LEFT">Multilevel</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'ML'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>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&nbsp;<A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>
for further details of the preconditioner.</TD>
</TR>
</TABLE>
</DIV></TD></TR>
</TABLE>
</DIV><P></P>
<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Note that the module <code>mld_prec_mod</code>, 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 <code>psb_base_mod</code>, for the sparse matrix and communication descriptor
data types, and <code>psb_krylov_mod</code>, for interfacing with the
Krylov solvers, must be also used (see Section&nbsp;<A HREF="node14.html#sec:examples">5.1</A>).
<BR></FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Remark 1.</B> 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. </FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><HR>
<!--Table of Child-Links-->
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
</OL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍ ␍In order to perform the coarsening step, the smoothed aggregation algorithm␍described in&nbsp;[<A
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>] is used. In this algorithm,␍each index <!-- MATH
$j \in \Omega^{k+1}$
-->
<IMG
WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img27.png"
ALT="$j \in \Omega^{k+1}$"> corresponds to an aggregate <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img28.png"
ALT="$\Omega^k_j$"> of <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$">,␍consisting of a suitably chosen index <!-- MATH
$i \in \Omega^k$
-->
<IMG
WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img29.png"
ALT="$i \in \Omega^k$"> and indices that are (usually) contained in a␍strongly-coupled neighborood of <IMG
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img30.png"
ALT="$i$">, i.e.,␍</FONT></FONT></FONT>
<BR>
<DIV ALIGN="RIGHT">
<UL>
<LI><A NAME="tex2html234"
HREF="node14.html">Examples</A>
</UL>
<!--End of Table of Child-Links-->
<HR>
<!-- MATH
\begin{equation}
\Omega^k_j \subset \mathcal{N}_i^k(\theta) = ␍ \left\{ r \in \Omega^k: |a_{ir}^k| > \theta \sqrt{|a_{ii}^ka_{rr}^k|} \right \} \cup \left\{ i \right\},
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:strongly_coup"></A><IMG
WIDTH="387" HEIGHT="72" BORDER="0"
SRC="img31.png"
ALT="\begin{displaymath}
\Omega^k_j \subset \mathcal{N}_i^k(\theta) = ␍ \left\{ r ...
...vert a_{ii}^ka_{rr}^k\vert} \right \} \cup \left\{ i \right\},␍\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(3)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍for a given threshold <!-- MATH
$\theta \in [0,1]$
-->
<IMG
WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img32.png"
ALT="$\theta \in [0,1]$"> (see&nbsp;[<A
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>] 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␍distribution. This version is embarrassingly parallel, since it does not require any data ␍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 <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$">. Nevertheless, this parallel algorithm has been chosen for␍MLD2P4, since it has been shown to produce good results in practice␍[<A
HREF="node29.html#aaecc_07">5</A>,<A
HREF="node29.html#apnum_07">7</A>,<A
HREF="node29.html#TUMINARO_TONG">24</A>].␍␍The prolongator <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"> is built starting from a tentative prolongator␍<!-- MATH
$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$
-->
<IMG
WIDTH="117" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img33.png"
ALT="$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$">, defined as␍</FONT></FONT></FONT>
<BR>
<DIV ALIGN="RIGHT">
<!-- MATH
\begin{equation}
\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.
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:tent_prol"></A><IMG
WIDTH="287" HEIGHT="51" BORDER="0"
SRC="img34.png"
ALT="\begin{displaymath}␍\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k = ␍\left\{...
...ega^k_j, \\ ␍0 &amp; \quad \mbox{otherwise},␍\end{array} \right.
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(4)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍where <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img28.png"
ALT="$\Omega^k_j$"> is the aggregate of <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$">␍corresponding to the index <!-- MATH
$j \in \Omega^{k+1}$
-->
<IMG
WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img27.png"
ALT="$j \in \Omega^{k+1}$">.␍<IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"> is obtained by applying to <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img35.png"
ALT="$\bar{P}^k$"> a smoother␍<!-- MATH
$S^k \in \mathbb{R}^{n_k \times n_k}$
-->
<IMG
WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img36.png"
ALT="$S^k \in \mathbb{R}^{n_k \times n_k}$">:␍</FONT></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
P^k = S^k \bar{P}^k,
\end{displaymath}
-->
<IMG
WIDTH="90" HEIGHT="30" BORDER="0"
SRC="img37.png"
ALT="\begin{displaymath}␍P^k = S^k \bar{P}^k,␍\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍in order to remove nonsmooth components from the range of the prolongator,␍and hence to improve the convergence properties of the multi-level␍method&nbsp;[<A
HREF="node29.html#BREZINA_VANEK">2</A>,<A
HREF="node29.html#Stuben_01">23</A>].␍A simple choice for <IMG
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img38.png"
ALT="$S^k$"> is the damped Jacobi smoother:␍</FONT></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
S^k = I - \omega^k (D^k)^{-1} A^k_F ,
\end{displaymath}
-->
<IMG
WIDTH="175" HEIGHT="31" BORDER="0"
SRC="img39.png"
ALT="\begin{displaymath}␍S^k = I - \omega^k (D^k)^{-1} A^k_F , ␍\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍where <IMG
WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img40.png"
ALT="$D^k$"> is the diagonal matrix with the same diagonal entries as <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$A^k$">,␍<!-- MATH
$A^k_F = (\bar{a}_{ij}^k)$
-->
<IMG
WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img42.png"
ALT="$A^k_F = (\bar{a}_{ij}^k)$"> is the filtered matrix defined as␍</FONT></FONT></FONT>
<BR>
<DIV ALIGN="RIGHT">
<!-- MATH
\begin{equation}
\bar{a}_{ij}^k =␍ \left \{ \begin{array}{ll}␍ a_{ij}^k & \mbox{if } j \in \mathcal{N}_i^k(\theta), \\␍ 0 & \mbox{otherwise},␍ \end{array} \right.␍ \; (j \ne i),␍ \qquad␍ \bar{a}_{ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:filtered"></A><IMG
WIDTH="514" HEIGHT="74" BORDER="0"
SRC="img43.png"
ALT="\begin{displaymath}
\bar{a}_{ij}^k =␍ \left \{ \begin{array}{ll}␍ a_{ij}^k &amp; ...
...ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),␍\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(5)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍and <IMG
WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img44.png"
ALT="$\omega^k$"> is an approximation of <IMG
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img45.png"
ALT="$4/(3\rho^k)$">, where␍<IMG
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img46.png"
ALT="$\rho^k$"> is the spectral radius of <!-- MATH
$(D^k)^{-1}A^k_F$
-->
<IMG
WIDTH="83" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img47.png"
ALT="$(D^k)^{-1}A^k_F$"> [<A
HREF="node29.html#BREZINA_VANEK">2</A>].␍In MLD2P4 this approximation is obtained by using <!-- MATH
$\| A^k_F \|_\infty$
-->
<IMG
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img48.png"
ALT="$\Vert A^k_F \Vert _\infty$"> as an estimate␍of <IMG
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img46.png"
ALT="$\rho^k$">. Note that for systems coming from uniformly elliptic␍problems, filtering the matrix <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$A^k$"> has little or no effect, and␍<IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$A^k$"> can be used instead of <IMG
WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img49.png"
ALT="$A^k_F$">. The latter choice is the default in MLD2P4.␍␍</FONT></FONT></FONT><HR>
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