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<H1><A NAME="SECTION00070000000000000000"></A><A NAME="sec:started"></A>
<BR>
Getting Started
</H1>
<P>
We describe the basics for building and applying MLD2P4 one-level and multi-level
Schwarz preconditioners with the Krylov solvers included in PSBLAS [<A
HREF="node25.html#PSBLASGUIDE">15</A>].
The following steps are required:
<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>
<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>mld_precinit</code>, which also sets defaults for each preconditioner
type selected by the user. The defaults associated to each preconditioner
type are given in Table&nbsp;<A HREF="#tab:precinit">1</A>, where the strings used by
<code>mld_precinit</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><I>Modify the selected preconditioner type, by properly setting
preconditioner parameters.</I> This is performed by the routine <code>mld_precset</code>.
This routine must be called only if the user wants to modify the default values
of the parameters associated to the selected preconditioner type, to obtain a variant
of the preconditioner. Examples of use of <code>mld_precset</code> are given in
Section&nbsp;<A HREF="node15.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="node16.html#sec:userinterface">6</A>, Tables&nbsp;<A HREF="#tab:p_type">2</A>-<A HREF="#tab:p_coarse">5</A>.
</LI>
<LI><I>Build the preconditioner for a given matrix.</I> This is performed by
the routine <code>mld_precbld</code>.
</LI>
<LI><I>Apply the preconditioner at each iteration of a Krylov solver.</I>
This is performed by the routine <code>mld_precaply</code>. When using the PSBLAS Krylov solvers,
this step is completely transparent to the user, since <code>mld_precaply</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>mld_</code> <code>precfree</code>. This step is complementary to step 1 and should
be performed when the preconditioner is no more used.
</LI>
</OL>
A detailed description of the above routines is given in Section&nbsp;<A HREF="node16.html#sec:userinterface">6</A>.
Examples showing the basic use of MLD2P4 are reported in Section&nbsp;<A HREF="node15.html#sec:examples">5.1</A>.
<P>
Note that the Fortran 95 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="node15.html#sec:examples">5.1</A>).
<P>
<BR><B>Remark 1.</B> The coarsest-level solver used by the default two-level
preconditioner has been chosen by taking into account that, on parallel
machines, it often leads to the smallest execution time when applied to
linear systems coming from finite-difference discretizations of basic
elliptic PDE problems, considered as standard tests for multi-level Schwarz
preconditioners [<A
HREF="node25.html#aaecc_07">3</A>,<A
HREF="node25.html#apnum_07">4</A>]. However, this solver does
not necessarily correspond to the smallest number of iterations of the
preconditioned Krylov method, which is usually obtained by applying
a direct solver to the coarsest-level system, e.g. based on the LU
factorization (see Section&nbsp;<A HREF="node16.html#sec:userinterface">6</A>
for the coarsest-level solvers available in MLD2P4).
<P>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="923"></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"><SMALL>STRING</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=221><SMALL>DEFAULT PRECONDITIONER</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT">No preconditioner</TD>
<TD ALIGN="LEFT"><code>'NOPREC'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=221>Considered only to use the PSBLAS
Krylov solvers with no preconditioner.</TD>
</TR>
<TR><TD ALIGN="LEFT">Diagonal</TD>
<TD ALIGN="LEFT"><code>'DIAG'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=221>--</TD>
</TR>
<TR><TD ALIGN="LEFT">Block Jacobi</TD>
<TD ALIGN="LEFT"><code>'BJAC'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=221>Block Jacobi with ILU(0) on the local blocks.</TD>
</TR>
<TR><TD ALIGN="LEFT">Additive Schwarz</TD>
<TD ALIGN="LEFT"><code>'AS'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=221>Restricted Additive Schwarz (RAS),
with overlap 1 and ILU(0) on the local blocks.</TD>
</TR>
<TR><TD ALIGN="LEFT">Multilevel</TD>
<TD ALIGN="LEFT"><code>'ML'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=221>Multi-level hybrid preconditioner (additive on the
same level and multiplicative through the levels),
with post-smoothing only.
Number of levels: 2.
Post-smoother: RAS with overlap 1 and ILU(0)
on the local blocks.
Aggregation: decoupled smoothed aggregation with
threshold <IMG
WIDTH="45" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img88.png"
ALT="$\theta = 0$">.
Coarsest matrix: distributed among the processors.
Coarsest-level solver:
4 sweeps of the block-Jacobi solver,
with LU or ILU factorization of the blocks
(UMFPACK for the double precision versions and
SuperLU for the single precision ones, if the packages
have been installed; ILU(0), otherwise).</TD>
</TR>
</TABLE>
</DIV>
<P>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR>
<P>
<BR><HR>
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