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<H1><A NAME="SECTION00070000000000000000"></A><A NAME="sec:started"></A>
<BR>
Getting Started
</H1><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
<P>
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">We describe the basics for building and applying MLD2P4 one-level and multilevel
(i.e., AMG) preconditioners with the Krylov solvers included in PSBLAS [<A
HREF="node36.html#PSBLASGUIDE">13</A>].
The following steps are required:
</BIG></BIG></BIG>
<OL>
<LI><SPAN CLASS="textit">Declare the preconditioner data structure</SPAN>. It is a derived data type,
<code>mld_</code><SPAN CLASS="textit">x</SPAN><code>prec_</code> <code>type</code>, where <SPAN CLASS="textit">x</SPAN> 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><SPAN CLASS="textit">Allocate and initialize the preconditioner data structure, according to
a preconditioner type chosen by the user</SPAN>. 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><SPAN CLASS="textit">Modify the selected preconditioner type, by properly setting
preconditioner parameters.</SPAN> 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="node17.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="node18.html#sec:userinterface">6</A>, Tables&nbsp;<A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>.
</LI>
<LI><SPAN CLASS="textit">Build the preconditioner for a given matrix</SPAN>. If the selected preconditioner
is multilevel, then two steps must be performed, as specified next.
<DL COMPACT>
<DT>4.1</DT>
<DD><SPAN CLASS="textit">Build the aggregation hierarchy for a given matrix.</SPAN> This is
performed by the routine <code>hierarchy_build</code>.
</DD>
<DT>4.2</DT>
<DD><SPAN CLASS="textit">Build the preconditioner for a given matrix.</SPAN> 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><SPAN CLASS="textit">Apply the preconditioner at each iteration of a Krylov solver.</SPAN>
This is performed by the method <code>apply</code>. When using the PSBLAS Krylov solvers,
this step is completely transparent to the user, since <code>apply</code> is called
by the PSBLAS routine implementing the Krylov solver (<code>psb_krylov</code>).
</LI>
<LI><SPAN CLASS="textit">Free the preconditioner data structure</SPAN>. 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><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
<P>
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">All the previous routines are available as methods of the preconditioner object.
A detailed description of them is given in Section&nbsp;<A HREF="node18.html#sec:userinterface">6</A>.
Examples showing the basic use of MLD2P4 are reported in Section&nbsp;<A HREF="node17.html#sec:examples">5.1</A>.
</BIG></BIG></BIG>
<P>
<BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="904"></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>'NONE'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Considered 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">Gauss-Seidel</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'GS'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Hybrid Gauss-Seidel (forward), that is,
global block Jacobi with
Gauss-Seidel as local solver.</TD>
</TR>
<TR><TD ALIGN="LEFT">Symmetrized Gauss-Seidel</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'FBGS'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Symmetrized hybrid Gauss-Seidel,that is,
forward Gauss-Seidel followed by
backward Gauss-Seidel.</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>Additive Schwarz (AS),
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><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
<P>
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">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="node17.html#sec:examples">5.1</A>).
<BR></BIG></BIG></BIG>
<P>
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"><SPAN CLASS="textbf">Remark 1.</SPAN> 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.
</BIG></BIG></BIG>
<P>
<BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
<BR><HR>
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