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original version by: Nikos Drakos, CBLU, University of Leeds
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<H1><A NAME="SECTION00070000000000000000"></A><A NAME="sec:started"></A>
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<BR>
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Getting Started
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</H1>
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<P>
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We describe the basics for building and applying MLD2P4 one-level and multi-level
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Schwarz preconditioners with the Krylov solvers included in PSBLAS [<A
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HREF="node25.html#PSBLASGUIDE">15</A>].
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The following steps are required:
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<OL>
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<LI><I>Declare the preconditioner data structure</I>. It is a derived data type,
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<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>
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or <code>z</code>, according to the basic data type of the sparse matrix
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(<code>s</code> = real single precision; <code>d</code> = real double precision;
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<code>c</code> = complex single precision; <code>z</code> = complex double precision).
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This data structure is accessed by the user only through the MLD2P4 routines,
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following an object-oriented approach.
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</LI>
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<LI><I>Allocate and initialize the preconditioner data structure, according to
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a preconditioner type chosen by the user</I>. This is performed by the routine
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<code>mld_precinit</code>, which also sets defaults for each preconditioner
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type selected by the user. The defaults associated to each preconditioner
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type are given in Table <A HREF="#tab:precinit">1</A>, where the strings used by
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<code>mld_precinit</code> to identify the preconditioner types are also given.
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Note that these strings are valid also if uppercase letters are substituted by
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corresponding lowercase ones.
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</LI>
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<LI><I>Modify the selected preconditioner type, by properly setting
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preconditioner parameters.</I> This is performed by the routine <code>mld_precset</code>.
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This routine must be called only if the user wants to modify the default values
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of the parameters associated to the selected preconditioner type, to obtain a variant
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of the preconditioner. Examples of use of <code>mld_precset</code> are given in
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Section <A HREF="node15.html#sec:examples">5.1</A>; a complete list of all the
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preconditioner parameters and their allowed and default values is provided in
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Section <A HREF="node16.html#sec:userinterface">6</A>, Tables <A HREF="#tab:p_type">2</A>-<A HREF="#tab:p_coarse">5</A>.
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</LI>
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<LI><I>Build the preconditioner for a given matrix.</I> This is performed by
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the routine <code>mld_precbld</code>.
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</LI>
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<LI><I>Apply the preconditioner at each iteration of a Krylov solver.</I>
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This is performed by the routine <code>mld_precaply</code>. When using the PSBLAS Krylov solvers,
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this step is completely transparent to the user, since <code>mld_precaply</code> is called
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by the PSBLAS routine implementing the Krylov solver (<code>psb_krylov</code>).
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</LI>
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<LI><I>Free the preconditioner data structure</I>. This is performed by
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the routine <code>mld_</code> <code>precfree</code>. This step is complementary to step 1 and should
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be performed when the preconditioner is no more used.
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</LI>
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</OL>
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A detailed description of the above routines is given in Section <A HREF="node16.html#sec:userinterface">6</A>.
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Examples showing the basic use of MLD2P4 are reported in Section <A HREF="node15.html#sec:examples">5.1</A>.
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<P>
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Note that the Fortran 95 module <code>mld_prec_mod</code>, containing the definition of the
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preconditioner data type and the interfaces to the routines of MLD2P4,
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must be used in any program calling such routines.
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The modules <code>psb_base_mod</code>, for the sparse matrix and communication descriptor
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data types, and <code>psb_krylov_mod</code>, for interfacing with the
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Krylov solvers, must be also used (see Section <A HREF="node15.html#sec:examples">5.1</A>).
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<P>
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<BR><B>Remark 1.</B> The coarsest-level solver used by the default two-level
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preconditioner has been chosen by taking into account that, on parallel
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machines, it often leads to the smallest execution time when applied to
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linear systems coming from finite-difference discretizations of basic
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elliptic PDE problems, considered as standard tests for multi-level Schwarz
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preconditioners [<A
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HREF="node25.html#aaecc_07">3</A>,<A
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HREF="node25.html#apnum_07">4</A>]. However, this solver does
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not necessarily correspond to the smallest number of iterations of the
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preconditioned Krylov method, which is usually obtained by applying
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a direct solver to the coarsest-level system, e.g. based on the LU
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factorization (see Section <A HREF="node16.html#sec:userinterface">6</A>
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for the coarsest-level solvers available in MLD2P4).
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<P>
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<BR><P></P>
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<DIV ALIGN="CENTER"><A NAME="924"></A>
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<TABLE>
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<CAPTION><STRONG>Table 1:</STRONG>
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Preconditioner types, corresponding strings and default choices.
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</CAPTION>
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<TR><TD>
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<DIV ALIGN="CENTER">
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<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
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<TR><TD ALIGN="LEFT"><SMALL>TYPE</SMALL></TD>
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<TD ALIGN="LEFT"><SMALL>STRING</SMALL></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=221><SMALL>DEFAULT PRECONDITIONER</SMALL></TD>
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</TR>
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<TR><TD ALIGN="LEFT">No preconditioner</TD>
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<TD ALIGN="LEFT"><code>'NOPREC'</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=221>Considered only to use the PSBLAS
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Krylov solvers with no preconditioner.</TD>
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</TR>
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<TR><TD ALIGN="LEFT">Diagonal</TD>
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<TD ALIGN="LEFT"><code>'DIAG'</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=221>--</TD>
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</TR>
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<TR><TD ALIGN="LEFT">Block Jacobi</TD>
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<TD ALIGN="LEFT"><code>'BJAC'</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=221>Block Jacobi with ILU(0) on the local blocks.</TD>
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</TR>
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<TR><TD ALIGN="LEFT">Additive Schwarz</TD>
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<TD ALIGN="LEFT"><code>'AS'</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=221>Restricted Additive Schwarz (RAS),
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with overlap 1 and ILU(0) on the local blocks.</TD>
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</TR>
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<TR><TD ALIGN="LEFT">Multilevel</TD>
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<TD ALIGN="LEFT"><code>'ML'</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=221>Multi-level hybrid preconditioner (additive on the
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same level and multiplicative through the levels),
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with pre- and post-smoothing.
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Number of levels: 2.
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Smoother: RAS with overlap 1 and ILU(0)
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on the local blocks.
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Aggregation: decoupled smoothed aggregation with
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threshold <IMG
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WIDTH="45" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img88.png"
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ALT="$\theta = 0$">.
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Coarsest matrix: distributed among the processors.
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Coarsest-level solver:
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4 sweeps of the block-Jacobi solver,
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with LU or ILU factorization of the blocks
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(UMFPACK for the double precision versions and
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SuperLU for the single precision ones, if the packages
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have been installed; ILU(0), otherwise).</TD>
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</TR>
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</TABLE>
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</DIV>
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<P>
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</TD></TR>
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</TABLE>
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</DIV><P></P>
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<BR>
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<P>
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<BR><HR>
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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