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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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<H2><A NAME="SECTION00062000000000000000"></A><A NAME="sec:aggregation"></A>
<H1><A NAME="SECTION00070000000000000000"></A><A NAME="sec:started"></A>
<BR>
Smoothed Aggregation
</H2>
Getting Started
</H1>
<P>
In order to define the restriction operator <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img69.png"
ALT="$R_C$">, which is used to compute
the coarse-level matrix <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img44.png"
ALT="$A_C$">, MLD2P4 uses the <I>smoothed aggregation</I>
algorithm described in [<A
HREF="node28.html#BREZINA_VANEK">1</A>,<A
HREF="node28.html#VANEK_MANDEL_BREZINA">27</A>].
The basic idea of this algorithm is to build a coarse set of vertices
<IMG
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img45.png"
ALT="$W_C$"> by suitably grouping the vertices of <IMG
WIDTH="23" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img11.png"
ALT="$W$"> into disjoint subsets
(aggregates), and to define the coarse-to-fine space transfer operator <IMG
WIDTH="29" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
SRC="img70.png"
ALT="$R_C^T$"> by
applying a suitable smoother to a simple piecewise constant
prolongation operator, to improve the quality of the coarse-space correction.
<P>
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 [<A
HREF="node27.html#PSBLASGUIDE">17</A>].
The following steps are required:
<OL>
<LI>coarsening of the vertex set <IMG
WIDTH="23" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img11.png"
ALT="$W$">, to obtain <IMG
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img45.png"
ALT="$W_C$">;
<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>
<LI>construction of the prolongator <IMG
WIDTH="29" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
SRC="img70.png"
ALT="$R_C^T$">;
<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>
<LI>application of <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img69.png"
ALT="$R_C$"> and <IMG
WIDTH="29" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
SRC="img70.png"
ALT="$R_C^T$"> to build <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img44.png"
ALT="$A_C$">.
<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>
<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>
<P>
To perform the coarsening step, we have implemented the aggregation algorithm sketched
in [<A
HREF="node28.html#apnum_07">4</A>]. According to [<A
HREF="node28.html#VANEK_MANDEL_BREZINA">27</A>], a modification of
this algorithm has been actually considered,
in which each aggregate <IMG
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img71.png"
ALT="$N_r$"> is made of vertices of <IMG
WIDTH="23" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img11.png"
ALT="$W$"> that are <I>strongly coupled</I>
to a certain root vertex <IMG
WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img72.png"
ALT="$r \in W$">, i.e. <BR><P></P>
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>.
<P>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="513"></A>
<TABLE>
<CAPTION><STRONG>Table 1:</STRONG>
Preconditioner types, corresponding strings and default choices.
</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
N_r = \left\{s \in W: |a_{rs}| > \theta \sqrt{|a_{rr}a_{ss}|} \right\}
\cup \left\{ r \right\} ,
\end{displaymath}
-->
<IMG
WIDTH="320" HEIGHT="38" BORDER="0"
SRC="img73.png"
ALT="\begin{displaymath}N_r = \left\{s \in W: \vert a_{rs}\vert &gt; \theta \sqrt{\vert a_{rr}a_{ss}\vert} \right\}
\cup \left\{ r \right\} ,
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
for a given <!-- MATH
$\theta \in [0,1]$
-->
<IMG
WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img74.png"
ALT="$\theta \in [0,1]$">.
Since this algorithm has a sequential nature, a <I>decoupled</I> version of
it has been chosen, where each processor <IMG
WIDTH="10" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img75.png"
ALT="$i$"> independently applies the algorithm to
the set of vertices <IMG
WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img76.png"
ALT="$W_i^0$"> 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 <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img2.png"
ALT="$A$">.
Nevertheless, this algorithm has been chosen for the implementation in MLD2P4,
since it has been shown to produce good results in practice
[<A
HREF="node28.html#aaecc_07">3</A>,<A
HREF="node28.html#apnum_07">4</A>,<A
HREF="node28.html#TUMINARO_TONG">26</A>].
<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 he 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>
<P>
The prolongator <IMG
WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
SRC="img77.png"
ALT="$P_C=R_C^T$"> is built starting from a <I>tentative prolongator</I>
<!-- MATH
$P \in \Re^{n \times n_C}$
-->
<IMG
WIDTH="90" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img78.png"
ALT="$P \in \Re^{n \times n_C}$">, defined as
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>
<DIV ALIGN="RIGHT">
<!-- MATH
\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. .
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:tent_prol"></A><IMG
WIDTH="291" HEIGHT="52" BORDER="0"
SRC="img79.png"
ALT="\begin{displaymath}
P=(p_{ij}), \quad p_{ij}=
\left\{ \begin{array}{ll}
1 &amp; \qu...
...\in V^j_C \\
0 &amp; \quad \mbox{otherwise}
\end{array} \right. .
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
<IMG
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img80.png"
ALT="$P_C$"> is obtained by
applying to <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img81.png"
ALT="$P$"> a smoother <!-- MATH
$S \in \Re^{n \times n}$
-->
<IMG
WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img82.png"
ALT="$S \in \Re^{n \times n}$">:
<BR>
<DIV ALIGN="RIGHT">
<!-- MATH
\begin{equation}
P_C = S P,
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:smoothed_prol"></A><IMG
WIDTH="73" HEIGHT="30" BORDER="0"
SRC="img83.png"
ALT="\begin{displaymath}
P_C = S P,
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(3)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
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 [<A
HREF="node28.html#BREZINA_VANEK">1</A>,<A
HREF="node28.html#Stuben_01">25</A>].
A simple choice for <IMG
WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img84.png"
ALT="$S$"> is the damped Jacobi smoother:
<BR>
<DIV ALIGN="RIGHT">
<!-- MATH
\begin{equation}
S = I - \omega D^{-1} A ,
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:jac_smoother"></A><IMG
WIDTH="125" HEIGHT="30" BORDER="0"
SRC="img85.png"
ALT="\begin{displaymath}
S = I - \omega D^{-1} A ,
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(4)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
where the value of <IMG
WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img86.png"
ALT="$\omega$"> can be chosen
using some estimate of the spectral radius of <IMG
WIDTH="50" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img87.png"
ALT="$D^{-1}A$"> [<A
HREF="node28.html#BREZINA_VANEK">1</A>].
<P>
<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.
<P>
<BR><HR>
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