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@@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<HTML>
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<HEAD>
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<TITLE>Getting Started</TITLE>
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<META NAME="description" CONTENT="Getting Started">
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<TITLE>Smoothed Aggregation</TITLE>
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<META NAME="description" CONTENT="Smoothed Aggregation">
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<META NAME="keywords" CONTENT="userhtml">
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<META NAME="resource-type" CONTENT="document">
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<META NAME="distribution" CONTENT="global">
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@@ -18,206 +18,314 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<LINK REL="STYLESHEET" HREF="userhtml.css">
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<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
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<BR>
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<B> Next:</B> <A NAME="tex2html233"
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HREF="node14.html">Examples</A>
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HREF="userhtml.html">userhtml</A>
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<B> Previous:</B> <A NAME="tex2html223"
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<B> Next:</B> <A NAME="tex2html242"
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HREF="node14.html">Smoothers and coarsest-level solvers</A>
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<B> Up:</B> <A NAME="tex2html238"
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HREF="node11.html">Multigrid Background</A>
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<B> Previous:</B> <A NAME="tex2html232"
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HREF="node12.html">AMG preconditioners</A>
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<B> <A NAME="tex2html231"
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<B> <A NAME="tex2html240"
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HREF="node2.html">Contents</A></B>
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<BR>
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<BR>
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<!--End of Navigation Panel-->
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<H1><A NAME="SECTION00070000000000000000"></A><A NAME="sec:started"></A>
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<H2><A NAME="SECTION00062000000000000000"></A><A NAME="sec:aggregation"></A>
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<BR>
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Getting Started
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</H1><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">We describe the basics for building and applying MLD2P4 one-level and multi-level
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(i.e., AMG) preconditioners with the Krylov solvers included in PSBLAS [<A
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HREF="node27.html#PSBLASGUIDE">13</A>].
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The following steps are required:
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</FONT></FONT></FONT>
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Smoothed Aggregation
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</H2><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍␍In order to define the prolongator <IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$">, used to compute␍the coarse-level matrix <IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img15.png"
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ALT="$A^{k+1}$">, MLD2P4 uses the smoothed aggregation␍algorithm described in [<A
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HREF="node29.html#BREZINA_VANEK">2</A>,<A
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HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>].␍The basic idea of this algorithm is to build a coarse set of indices␍<IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img26.png"
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ALT="$\Omega^{k+1}$"> by suitably grouping the indices of <IMG
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WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\Omega^k$"> into disjoint␍subsets (aggregates), and to define the coarse-to-fine space transfer operator␍<IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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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>
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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>aggregation of the indices of <IMG
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WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\Omega^k$"> to obtain <IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img26.png"
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ALT="$\Omega^{k+1}$">;
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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>init</code>, which also sets defaults for each preconditioner
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type selected by the user. The preconditioner types and the defaults associated
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with them are given in Table <A HREF="#tab:precinit">1</A>, where the strings used by
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<code>init</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>construction of the prolongator <IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$">;
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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>set</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 with the selected preconditioner type, to obtain a variant
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of that preconditioner. Examples of use of <code>set</code> are given in
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Section <A HREF="node14.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="node15.html#sec:userinterface">6</A>, Tables <A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>.
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<LI>application of <IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$"> and <IMG
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WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img17.png"
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ALT="$R^k=(P^k)^T$"> to build <IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img15.png"
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ALT="$A^{k+1}$">.
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</LI>
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<LI><I>Build the preconditioner for a given matrix</I>. If the selected preconditioner
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is multi-level, then two steps must be performed, as specified next.
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<DL COMPACT>
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<DT>4.1</DT>
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<DD><I>Build the aggregation hierarchy for a given matrix.</I> This is
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performed by the routine <code>hierarchy_build</code>.
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</DD>
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<DT>4.2</DT>
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<DD><I>Build the preconditioner for a given matrix.</I> This is performed
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by the routine <code>smoothers_build</code>.
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</DD>
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</DL>
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If the selected preconditioner is one-level, it is built in a single step,
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performed by the routine <code>bld</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>aply</code>. When using the PSBLAS Krylov solvers,
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this step is completely transparent to the user, since <code>aply</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>free</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><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">All the previous routines are available as methods of the preconditioner object.
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A detailed description of them is given in Section <A HREF="node15.html#sec:userinterface">6</A>.
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Examples showing the basic use of MLD2P4 are reported in Section <A HREF="node14.html#sec:examples">5.1</A>.
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</FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<BR><P></P>
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<DIV ALIGN="CENTER"><A NAME="532"></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" VALIGN="TOP" WIDTH=51><SMALL>STRING</SMALL></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232><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" VALIGN="TOP" WIDTH=51><code>'NOPREC'</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>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" VALIGN="TOP" WIDTH=51><code>'DIAG'</code> or <code>'JACOBI'</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Diagonal preconditioner.
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For any zero diagonal entry of the matrix to be preconditioned,
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the corresponding entry of the preconditioner is set to 1.</TD>
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</TR>
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<TR><TD ALIGN="LEFT">Block Jacobi</TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'BJAC'</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>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" VALIGN="TOP" WIDTH=51><code>'AS'</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>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" VALIGN="TOP" WIDTH=51><code>'ML'</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>V-cycle with one hybrid forward Gauss-Seidel
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(GS) sweep as pre-smoother and one hybrid backward
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GS sweep as post-smoother, basic smoothed aggregation
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as coarsening algorithm, and LU (plus triangular solve)
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as coarsest-level solver. See the default values in
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Tables <A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>
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for further details of the preconditioner.</TD>
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</TR>
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</TABLE>
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</DIV></TD></TR>
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</TABLE>
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</DIV><P></P>
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<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Note that the 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="node14.html#sec:examples">5.1</A>).
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<BR></FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Remark 1.</B> Coarsest-level solvers based on the LU factorization,
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such as those implemented in UMFPACK, MUMPS, SuperLU, and SuperLU_Dist,
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usually lead to smaller numbers of preconditioned Krylov
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iterations than inexact solvers, when the linear system comes from
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a standard discretization of basic scalar elliptic PDE problems. However,
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this does not necessarily correspond to the smallest execution time
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on parallel computers. </FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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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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</OL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍ ␍In order to perform the coarsening step, the smoothed aggregation algorithm␍described in [<A
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HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>] is used. In this algorithm,␍each index <!-- MATH
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$j \in \Omega^{k+1}$
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-->
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<IMG
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WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img27.png"
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ALT="$j \in \Omega^{k+1}$"> corresponds to an aggregate <IMG
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WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img28.png"
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ALT="$\Omega^k_j$"> of <IMG
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WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\Omega^k$">,␍consisting of a suitably chosen index <!-- MATH
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$i \in \Omega^k$
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-->
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<IMG
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WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img29.png"
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ALT="$i \in \Omega^k$"> and indices that are (usually) contained in a␍strongly-coupled neighborood of <IMG
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WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img30.png"
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ALT="$i$">, i.e.,␍</FONT></FONT></FONT>
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<BR>
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<DIV ALIGN="RIGHT">
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<UL>
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<LI><A NAME="tex2html234"
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HREF="node14.html">Examples</A>
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</UL>
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<!--End of Table of Child-Links-->
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<HR>
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<!-- MATH
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\begin{equation}
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\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\},
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\end{equation}
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-->
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<TABLE WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:strongly_coup"></A><IMG
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WIDTH="387" HEIGHT="72" BORDER="0"
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SRC="img31.png"
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ALT="\begin{displaymath}
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\Omega^k_j \subset \mathcal{N}_i^k(\theta) = ␍ \left\{ r ...
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...vert a_{ii}^ka_{rr}^k\vert} \right \} \cup \left\{ i \right\},␍\end{displaymath}"></TD>
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<TD WIDTH=10 ALIGN="RIGHT">
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(3)</TD></TR>
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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍for a given threshold <!-- MATH
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$\theta \in [0,1]$
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-->
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<IMG
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WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img32.png"
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ALT="$\theta \in [0,1]$"> (see [<A
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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
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$A$">. Nevertheless, this parallel algorithm has been chosen for␍MLD2P4, since it has been shown to produce good results in practice␍[<A
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HREF="node29.html#aaecc_07">5</A>,<A
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HREF="node29.html#apnum_07">7</A>,<A
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HREF="node29.html#TUMINARO_TONG">24</A>].␍␍The prolongator <IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$"> is built starting from a tentative prolongator␍<!-- MATH
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$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$
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-->
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<IMG
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WIDTH="117" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img33.png"
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ALT="$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$">, defined as␍</FONT></FONT></FONT>
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<BR>
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<DIV ALIGN="RIGHT">
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<!-- MATH
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\begin{equation}
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\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.
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\end{equation}
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-->
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<TABLE WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:tent_prol"></A><IMG
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WIDTH="287" HEIGHT="51" BORDER="0"
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SRC="img34.png"
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ALT="\begin{displaymath}␍\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k = ␍\left\{...
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...ega^k_j, \\ ␍0 & \quad \mbox{otherwise},␍\end{array} \right.
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\end{displaymath}"></TD>
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<TD WIDTH=10 ALIGN="RIGHT">
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(4)</TD></TR>
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</TABLE>
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<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 [<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 & ...
|
||||
...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>
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html232"
|
||||
<A NAME="tex2html241"
|
||||
HREF="node14.html">
|
||||
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A>
|
||||
<A NAME="tex2html228"
|
||||
HREF="userhtml.html">
|
||||
<A NAME="tex2html237"
|
||||
HREF="node11.html">
|
||||
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A>
|
||||
<A NAME="tex2html222"
|
||||
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HREF="node12.html">
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<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A>
|
||||
<A NAME="tex2html230"
|
||||
<A NAME="tex2html239"
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||||
HREF="node2.html">
|
||||
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
|
||||
<BR>
|
||||
<B> Next:</B> <A NAME="tex2html233"
|
||||
HREF="node14.html">Examples</A>
|
||||
<B> Up:</B> <A NAME="tex2html229"
|
||||
HREF="userhtml.html">userhtml</A>
|
||||
<B> Previous:</B> <A NAME="tex2html223"
|
||||
<B> Next:</B> <A NAME="tex2html242"
|
||||
HREF="node14.html">Smoothers and coarsest-level solvers</A>
|
||||
<B> Up:</B> <A NAME="tex2html238"
|
||||
HREF="node11.html">Multigrid Background</A>
|
||||
<B> Previous:</B> <A NAME="tex2html232"
|
||||
HREF="node12.html">AMG preconditioners</A>
|
||||
<B> <A NAME="tex2html231"
|
||||
<B> <A NAME="tex2html240"
|
||||
HREF="node2.html">Contents</A></B>
|
||||
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|
||||
|
||||
|
||||
Reference in New Issue
Block a user