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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>Smoothed Aggregation</TITLE>
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<META NAME="description" CONTENT="Smoothed Aggregation">
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<TITLE>Multi-level Schwarz Preconditioners</TITLE>
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<META NAME="description" CONTENT="Multi-level Schwarz Preconditioners">
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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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@@ -18,306 +18,751 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<BODY >
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<BR>
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<B> Next:</B> <A NAME="tex2html205"
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HREF="node13.html">Getting Started</A>
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<B> Up:</B> <A NAME="tex2html201"
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HREF="node10.html">Multi-level Domain Decomposition Background</A>
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<B> Previous:</B> <A NAME="tex2html197"
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HREF="node11.html">Multi-level Schwarz Preconditioners</A>
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<B> <A NAME="tex2html203"
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<B> Next:</B> <A NAME="tex2html210"
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HREF="node13.html">Smoothed Aggregation</A>
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<B> Up:</B> <A NAME="tex2html206"
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HREF="node11.html">Multi-level Domain Decomposition Background</A>
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<B> Previous:</B> <A NAME="tex2html200"
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HREF="node11.html">Multi-level Domain Decomposition Background</A>
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<B> <A NAME="tex2html208"
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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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<H2><A NAME="SECTION00062000000000000000"></A><A NAME="sec:aggregation"></A>
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<H2><A NAME="SECTION00061000000000000000"></A><A NAME="sec:multilevel"></A>
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<BR>
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Smoothed Aggregation
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Multi-level Schwarz Preconditioners
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</H2>
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<P>
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In order to define the restriction operator <IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img67.png"
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ALT="$R_C$">, which is used to compute
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the coarse-level matrix <IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img43.png"
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ALT="$A_C$">, MLD2P4 uses the <I>smoothed aggregation</I>
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algorithm described in [<A
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HREF="node24.html#BREZINA_VANEK">1</A>,<A
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HREF="node24.html#VANEK_MANDEL_BREZINA">24</A>].
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The basic idea of this algorithm is to build a coarse set of vertices
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The Multilevel preconditioners implemented in MLD2P4 are obtained by combining
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AS preconditioners with coarse-space corrections; therefore
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we first provide a sketch of the AS preconditioners.
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<P>
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Given the linear system ,
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where <!-- MATH
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$A=(a_{ij}) \in \Re^{n \times n}$
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-->
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<IMG
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WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img44.png"
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ALT="$W_C$"> by suitably grouping the vertices of <IMG
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WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
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SRC="img3.png"
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ALT="$A=(a_{ij}) \in \Re^{n \times n}$"> is a
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nonsingular sparse matrix with a symmetric nonzero pattern,
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let <IMG
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WIDTH="93" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img4.png"
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ALT="$G=(W,E)$"> be the adjacency graph of <IMG
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img2.png"
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ALT="$A$">, where <!-- MATH
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$W=\{1, 2, \ldots, n\}$
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-->
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<IMG
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WIDTH="138" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.png"
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ALT="$W=\{1, 2, \ldots, n\}$">
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and <!-- MATH
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$E=\{(i,j) : a_{ij} \neq 0\}$
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-->
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<IMG
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WIDTH="162" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img6.png"
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ALT="$E=\{(i,j) : a_{ij} \neq 0\}$"> are the vertex set and the edge set of <IMG
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img7.png"
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ALT="$G$">,
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respectively. Two vertices are called adjacent if there is an edge connecting
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them. For any integer <IMG
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WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img8.png"
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ALT="$\delta > 0$">, a <IMG
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WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\delta$">-overlap
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partition of <IMG
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WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img10.png"
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ALT="$W$"> into disjoint subsets
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(aggregates), and to define the coarse-to-fine space transfer operator <IMG
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WIDTH="29" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
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SRC="img68.png"
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ALT="$R_C^T$"> by
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applying a suitable smoother to a simple piecewise constant
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prolongation operator, to improve the quality of the coarse-space correction.
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ALT="$W$"> can be defined recursively as follows.
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Given a 0-overlap (or non-overlapping) partition of <IMG
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WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img10.png"
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ALT="$W$">,
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i.e. a set of <IMG
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WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img11.png"
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ALT="$m$"> disjoint nonempty sets <!-- MATH
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$W_i^0 \subset W$
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-->
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<IMG
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WIDTH="73" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img12.png"
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ALT="$W_i^0 \subset W$"> such that
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<!-- MATH
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$\cup_{i=1}^m W_i^0 = W$
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-->
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<IMG
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WIDTH="107" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img13.png"
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ALT="$\cup_{i=1}^m W_i^0 = W$">, a <IMG
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WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\delta$">-overlap
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partition of <IMG
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WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img10.png"
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ALT="$W$"> is obtained by considering the sets
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<!-- MATH
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$W_i^\delta \supset W_i^{\delta-1}$
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-->
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<IMG
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WIDTH="97" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
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SRC="img14.png"
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ALT="$W_i^\delta \supset W_i^{\delta-1}$"> obtained by including the vertices that
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are adjacent to any vertex in <!-- MATH
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$W_i^{\delta-1}$
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-->
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<IMG
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WIDTH="48" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
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SRC="img15.png"
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ALT="$W_i^{\delta-1}$">.
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<P>
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Three main steps can be identified in the smoothed aggregation procedure:
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<OL>
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<LI>coarsening of the vertex set <IMG
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WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img10.png"
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ALT="$W$">, to obtain <IMG
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WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img44.png"
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ALT="$W_C$">;
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</LI>
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<LI>construction of the prolongator <IMG
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WIDTH="29" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
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SRC="img68.png"
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ALT="$R_C^T$">;
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</LI>
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<LI>application of <IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img67.png"
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ALT="$R_C$"> and <IMG
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WIDTH="29" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
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SRC="img68.png"
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ALT="$R_C^T$"> to build <IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img43.png"
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ALT="$A_C$">.
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</LI>
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</OL>
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Let <IMG
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WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img16.png"
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ALT="$n_i^\delta$"> be the size of <IMG
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WIDTH="30" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img17.png"
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ALT="$W_i^\delta$"> and <!-- MATH
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$R_i^{\delta} \in
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\Re^{n_i^\delta \times n}$
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-->
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<IMG
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WIDTH="93" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
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SRC="img18.png"
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ALT="$R_i^{\delta} \in
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\Re^{n_i^\delta \times n}$"> the restriction operator that maps
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a vector <IMG
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WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img19.png"
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ALT="$v \in \Re^n$"> onto the vector <!-- MATH
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||||
$v_i^{\delta} \in \Re^{n_i^\delta}$
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-->
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<IMG
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WIDTH="70" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
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||||
SRC="img20.png"
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ALT="$v_i^{\delta} \in \Re^{n_i^\delta}$">
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containing the components of <IMG
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WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$v$"> corresponding to the vertices in
|
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<IMG
|
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WIDTH="30" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img17.png"
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ALT="$W_i^\delta$">. The transpose of <IMG
|
||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img22.png"
|
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ALT="$R_i^{\delta}$"> is a
|
||||
prolongation operator from <!-- MATH
|
||||
$\Re^{n_i^\delta}$
|
||||
-->
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<IMG
|
||||
WIDTH="32" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
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SRC="img23.png"
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ALT="$\Re^{n_i^\delta}$"> to <IMG
|
||||
WIDTH="26" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
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SRC="img24.png"
|
||||
ALT="$\Re^n$">.
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The matrix <!-- MATH
|
||||
$A_i^\delta=R_i^\delta A (R_i^\delta)^T \in
|
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\Re^{n_i^\delta \times n_i^\delta}$
|
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-->
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<IMG
|
||||
WIDTH="201" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img25.png"
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ALT="$A_i^\delta=R_i^\delta A (R_i^\delta)^T \in
|
||||
\Re^{n_i^\delta \times n_i^\delta}$"> can be considered
|
||||
as a restriction of <IMG
|
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img2.png"
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ALT="$A$"> corresponding to the set <IMG
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||||
WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img26.png"
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ALT="$W_i^{\delta}$">.
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<P>
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To perform the coarsening step, we have implemented the aggregation algorithm sketched
|
||||
in [<A
|
||||
HREF="node24.html#apnum_07">4</A>]. According to [<A
|
||||
HREF="node24.html#VANEK_MANDEL_BREZINA">24</A>], a modification of
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||||
this algorithm has been actually considered,
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in which each aggregate <IMG
|
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WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img69.png"
|
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ALT="$N_r$"> is made of vertices of <IMG
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WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img10.png"
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ALT="$W$"> that are <I>strongly coupled</I>
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to a certain root vertex <IMG
|
||||
WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img70.png"
|
||||
ALT="$r \in W$">, i.e. <BR><P></P>
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The <I>classical one-level AS</I> preconditioner is defined by
|
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<BR><P></P>
|
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<DIV ALIGN="CENTER">
|
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<!-- MATH
|
||||
\begin{displaymath}
|
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N_r = \left\{s \in W: |a_{rs}| > \theta \sqrt{|a_{rr}a_{ss}|} \right\}
|
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\cup \left\{ r \right\} ,
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M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
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(A_i^\delta)^{-1} R_i^{\delta},
|
||||
\end{displaymath}
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-->
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||||
|
||||
<IMG
|
||||
WIDTH="319" HEIGHT="38" BORDER="0"
|
||||
SRC="img71.png"
|
||||
ALT="\begin{displaymath}N_r = \left\{s \in W: \vert a_{rs}\vert > \theta \sqrt{\vert a_{rr}a_{ss}\vert} \right\}
|
||||
\cup \left\{ r \right\} ,
|
||||
WIDTH="206" HEIGHT="58" BORDER="0"
|
||||
SRC="img27.png"
|
||||
ALT="\begin{displaymath}
|
||||
M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
||||
(A_i^\delta)^{-1} R_i^{\delta},
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
for a given <!-- MATH
|
||||
$\theta \in [0,1]$
|
||||
where <IMG
|
||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img28.png"
|
||||
ALT="$A_i^\delta$"> is assumed to be nonsingular. Its application
|
||||
to a vector <IMG
|
||||
WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img19.png"
|
||||
ALT="$v \in \Re^n$"> within a Krylov solver requires the following
|
||||
three steps:
|
||||
|
||||
<OL>
|
||||
<LI>restriction of <IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$"> as <!-- MATH
|
||||
$v_i = R_i^{\delta} v$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img72.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="img73.png"
|
||||
ALT="$i$"> independently applies the algorithm to
|
||||
the set of vertices <IMG
|
||||
WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img74.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="node24.html#aaecc_07">3</A>,<A
|
||||
HREF="node24.html#apnum_07">4</A>,<A
|
||||
HREF="node24.html#TUMINARO_TONG">23</A>].
|
||||
WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img29.png"
|
||||
ALT="$v_i = R_i^{\delta} v$">, <IMG
|
||||
WIDTH="97" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img30.png"
|
||||
ALT="$i=1,\ldots,m$">;
|
||||
</LI>
|
||||
<LI>solution of the linear systems <!-- MATH
|
||||
$A_i^\delta w_i = v_i$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="80" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img31.png"
|
||||
ALT="$A_i^\delta w_i = v_i$">,
|
||||
<IMG
|
||||
WIDTH="97" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img30.png"
|
||||
ALT="$i=1,\ldots,m$">;
|
||||
</LI>
|
||||
<LI>prolongation and sum of the <IMG
|
||||
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img32.png"
|
||||
ALT="$w_i$">'s, i.e. <!-- MATH
|
||||
$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="144" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img33.png"
|
||||
ALT="$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$">.
|
||||
</LI>
|
||||
</OL>
|
||||
Note that the linear systems at step 2 are usually solved approximately,
|
||||
e.g. using incomplete LU factorizations such as ILU(<IMG
|
||||
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img34.png"
|
||||
ALT="$p$">), MILU(<IMG
|
||||
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img34.png"
|
||||
ALT="$p$">) and
|
||||
ILU(<IMG
|
||||
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img35.png"
|
||||
ALT="$p,t$">) [<A
|
||||
HREF="node25.html#Saad_book">19</A>, Chapter 10].
|
||||
|
||||
<P>
|
||||
The prolongator <IMG
|
||||
WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img75.png"
|
||||
ALT="$P_C=R_C^T$"> is built starting from a <I>tentative prolongator</I>
|
||||
<!-- MATH
|
||||
$P \in \Re^{n \times n_C}$
|
||||
A variant of the classical AS preconditioner that outperforms it
|
||||
in terms of convergence rate and of computation and communication
|
||||
time on parallel distributed-memory computers is the so-called <I>Restricted AS
|
||||
(RAS)</I> preconditioner [<A
|
||||
HREF="node25.html#CAI_SARKIS">5</A>,<A
|
||||
HREF="node25.html#EFSTATHIOU">13</A>]. It
|
||||
is obtained by zeroing the components of <IMG
|
||||
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img32.png"
|
||||
ALT="$w_i$"> corresponding to the
|
||||
overlapping vertices when applying the prolongation. Therefore,
|
||||
RAS differs from classical AS by the prolongation operators,
|
||||
which are substituted by <!-- MATH
|
||||
$(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="90" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img76.png"
|
||||
ALT="$P \in \Re^{n \times n_C}$">, defined as
|
||||
<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}
|
||||
WIDTH="118" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img36.png"
|
||||
ALT="$(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$">,
|
||||
where <IMG
|
||||
WIDTH="25" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img37.png"
|
||||
ALT="$\tilde{R}_i^0$"> is obtained by zeroing the rows of <IMG
|
||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img38.png"
|
||||
ALT="$R_i^\delta$">
|
||||
corresponding to the vertices in <!-- MATH
|
||||
$W_i^\delta \backslash W_i^0$
|
||||
-->
|
||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:tent_prol"></A><IMG
|
||||
WIDTH="290" HEIGHT="52" BORDER="0"
|
||||
SRC="img77.png"
|
||||
<IMG
|
||||
WIDTH="66" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img39.png"
|
||||
ALT="$W_i^\delta \backslash W_i^0$">:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
|
||||
(A_i^\delta)^{-1} R_i^{\delta}.
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="216" HEIGHT="58" BORDER="0"
|
||||
SRC="img40.png"
|
||||
ALT="\begin{displaymath}
|
||||
P=(p_{ij}), \quad p_{ij}=
|
||||
\left\{ \begin{array}{ll}
|
||||
1 & \qu...
|
||||
...\in V^j_C \\
|
||||
0 & \quad \mbox{otherwise}
|
||||
\end{array} \right. .
|
||||
\end{displaymath}"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
(2)</TD></TR>
|
||||
</TABLE>
|
||||
<BR CLEAR="ALL"></DIV><P></P>
|
||||
M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
|
||||
(A_i^\delta)^{-1} R_i^{\delta}.
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
Analogously, the AS variant called <I>AS with Harmonic extension (ASH)</I>
|
||||
is defined by
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
||||
(A_i^\delta)^{-1} \tilde{R}_i^0.
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="218" HEIGHT="58" BORDER="0"
|
||||
SRC="img41.png"
|
||||
ALT="\begin{displaymath}M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
||||
(A_i^\delta)^{-1} \tilde{R}_i^0.
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
We note that for <IMG
|
||||
WIDTH="45" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img42.png"
|
||||
ALT="$\delta=0$"> the three variants of the AS preconditioner are
|
||||
all equal to the block-Jacobi preconditioner.
|
||||
|
||||
<P>
|
||||
As already observed, the convergence rate of the one-level Schwarz
|
||||
preconditioned iterative solvers deteriorates as the number <IMG
|
||||
WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img11.png"
|
||||
ALT="$m$"> of partitions
|
||||
of <IMG
|
||||
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img10.png"
|
||||
ALT="$W$"> increases [<A
|
||||
HREF="node25.html#dd1_94">7</A>,<A
|
||||
HREF="node25.html#dd2_96">20</A>]. To reduce the dependency
|
||||
of the number of iterations on the degree of parallelism we may
|
||||
introduce a global coupling among the overlapping partitions by defining
|
||||
a coarse-space approximation <IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="$A_C$"> of the matrix <IMG
|
||||
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$A$">.
|
||||
In a pure algebraic setting, <IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="$A_C$"> is usually built with
|
||||
a Galerkin approach. Given a set <IMG
|
||||
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img44.png"
|
||||
ALT="$W_C$"> of <I>coarse vertices</I>,
|
||||
with size <IMG
|
||||
WIDTH="26" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img45.png"
|
||||
ALT="$n_C$">, and a suitable restriction operator
|
||||
<!-- MATH
|
||||
$R_C \in \Re^{n_C \times n}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="101" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img46.png"
|
||||
ALT="$R_C \in \Re^{n_C \times n}$">, <IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="$A_C$"> is defined as
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
A_C=R_C A R_C^T
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="109" HEIGHT="31" BORDER="0"
|
||||
SRC="img47.png"
|
||||
ALT="\begin{displaymath}
|
||||
A_C=R_C A R_C^T
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
and the coarse-level correction matrix to be combined with a generic
|
||||
one-level AS preconditioner <IMG
|
||||
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img48.png"
|
||||
ALT="$M_{1L}$"> is obtained as
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{C}^{-1}= R_C^T A_C^{-1} R_C,
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="144" HEIGHT="32" BORDER="0"
|
||||
SRC="img49.png"
|
||||
ALT="\begin{displaymath}
|
||||
M_{C}^{-1}= R_C^T A_C^{-1} R_C,
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
where <IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="$A_C$"> is assumed to be nonsingular. The application of <IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img50.png"
|
||||
ALT="$M_{C}^{-1}$">
|
||||
to a vector <IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$"> corresponds to a restriction, a solution and
|
||||
a prolongation step; the solution step, involving the matrix <IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="$A_C$">,
|
||||
may be carried out also approximately.
|
||||
|
||||
<P>
|
||||
The combination of <IMG
|
||||
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img51.png"
|
||||
ALT="$M_{C}$"> and <IMG
|
||||
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img48.png"
|
||||
ALT="$M_{1L}$"> may be
|
||||
performed in either an additive or a multiplicative framework.
|
||||
In the former case, the <I>two-level additive</I> Schwarz preconditioner
|
||||
is obtained:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="166" HEIGHT="32" BORDER="0"
|
||||
SRC="img52.png"
|
||||
ALT="\begin{displaymath}
|
||||
M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
Applying <IMG
|
||||
WIDTH="59" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img53.png"
|
||||
ALT="$M_{2L-A}^{-1}$"> to a vector <IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$"> within a Krylov solver
|
||||
corresponds to applying <IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img50.png"
|
||||
ALT="$M_{C}^{-1}$">
|
||||
and <IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img54.png"
|
||||
ALT="$M_{1L}^{-1}$"> to <IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$"> independently and then summing up
|
||||
the results.
|
||||
|
||||
<P>
|
||||
In the multiplicative case, the combination can be
|
||||
performed by first applying the smoother <IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img54.png"
|
||||
ALT="$M_{1L}^{-1}$"> and then
|
||||
the coarse-level correction operator <IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img50.png"
|
||||
ALT="$M_{C}^{-1}$">:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
\begin{array}{l}
|
||||
w = M_{1L}^{-1} v, \\
|
||||
z = w + M_{C}^{-1} (v-Aw);
|
||||
\end{array}
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="177" HEIGHT="51" BORDER="0"
|
||||
SRC="img55.png"
|
||||
ALT="\begin{displaymath}
|
||||
\begin{array}{l}
|
||||
w = M_{1L}^{-1} v, \\
|
||||
z = w + M_{C}^{-1} (v-Aw);
|
||||
\end{array}\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
this corresponds to the following <I>two-level hybrid pre-smoothed</I>
|
||||
Schwarz preconditioner:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="308" HEIGHT="33" BORDER="0"
|
||||
SRC="img56.png"
|
||||
ALT="\begin{displaymath}
|
||||
M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
On the other hand, by applying the smoother after the coarse-level correction,
|
||||
i.e. by computing
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
\begin{array}{l}
|
||||
w = M_{C}^{-1} v , \\
|
||||
z = w + M_{1L}^{-1} (v-Aw) ,
|
||||
\end{array}
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="177" HEIGHT="51" BORDER="0"
|
||||
SRC="img57.png"
|
||||
ALT="\begin{displaymath}
|
||||
\begin{array}{l}
|
||||
w = M_{C}^{-1} v , \\
|
||||
z = w + M_{1L}^{-1} (v-Aw) ,
|
||||
\end{array}\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
the <I>two-level hybrid post-smoothed</I>
|
||||
Schwarz preconditioner is obtained:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="317" HEIGHT="33" BORDER="0"
|
||||
SRC="img58.png"
|
||||
ALT="\begin{displaymath}
|
||||
M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
One more variant of two-level hybrid preconditioner is obtained by applying
|
||||
the smoother before and after the coarse-level correction. In this case, the
|
||||
preconditioner is symmetric if <IMG
|
||||
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$A$">, <IMG
|
||||
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img48.png"
|
||||
ALT="$M_{1L}$"> and <IMG
|
||||
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img51.png"
|
||||
ALT="$M_{C}$"> are symmetric.
|
||||
|
||||
<P>
|
||||
As previously noted, on parallel computers the number of submatrices usually matches
|
||||
the number of available processors. When the size of the system to be preconditioned
|
||||
is very large, the use of many processors, i.e. of many small submatrices, often
|
||||
leads to a large coarse-level system, whose solution may be computationally expensive.
|
||||
On the other hand, the use of few processors often leads to local sumatrices that
|
||||
are too expensive to be processed on single processors, because of memory and/or
|
||||
computing requirements. Therefore, it seems natural to use a recursive approach,
|
||||
in which the coarse-level correction is re-applied starting from the current
|
||||
coarse-level system. The corresponding preconditioners, called <I>multi-level</I>
|
||||
preconditioners, can significantly reduce the computational cost of preconditioning
|
||||
with respect to the two-level case (see [<A
|
||||
HREF="node25.html#dd2_96">20</A>, Chapter 3]).
|
||||
Additive and hybrid multilevel preconditioners
|
||||
are obtained as direct extensions of the two-level counterparts.
|
||||
For a detailed descrition of them, the reader is
|
||||
referred to [<A
|
||||
HREF="node25.html#dd2_96">20</A>, Chapter 3].
|
||||
The algorithm for the application of a multi-level hybrid
|
||||
post-smoothed preconditioner <IMG
|
||||
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img59.png"
|
||||
ALT="$M$"> to a vector <IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$">, i.e. for the
|
||||
computation of <IMG
|
||||
WIDTH="87" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img60.png"
|
||||
ALT="$w=M^{-1}v$">, is reported, for
|
||||
example, in Figure <A HREF="#fig:mlhpost_alg">1</A>. Here the number of levels
|
||||
is denoted by <IMG
|
||||
WIDTH="37" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img61.png"
|
||||
ALT="$nlev$"> and the levels are numbered in increasing order starting
|
||||
from the finest one, i.e. the finest level is level 1; the coarse matrix
|
||||
and the corresponding basic preconditioner at each level <IMG
|
||||
WIDTH="10" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img62.png"
|
||||
ALT="$l$"> are denoted by <IMG
|
||||
WIDTH="22" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img63.png"
|
||||
ALT="$A_l$"> and
|
||||
<IMG
|
||||
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img78.png"
|
||||
ALT="$P_C$"> is obtained by
|
||||
applying to <IMG
|
||||
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img79.png"
|
||||
ALT="$P$"> a smoother <!-- MATH
|
||||
$S \in \Re^{n \times n}$
|
||||
SRC="img64.png"
|
||||
ALT="$M_l$">, respectively, with <IMG
|
||||
WIDTH="61" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img65.png"
|
||||
ALT="$A_1=A$">.
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:mlhpost_alg"></A><A NAME="508"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
|
||||
Application of the multi-level hybrid post-smoothed preconditioner.</CAPTION>
|
||||
<TR><TD>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
$\framebox{
|
||||
\begin{minipage}{.85\textwidth} {\small
|
||||
\begin{tabbing}
|
||||
\quad \=\quad \=\quad \=\quad \\[-1mm]
|
||||
$v_1 = v$; \\[2mm]
|
||||
\textbf{for $l=2, nlev$\ do}\\[1mm]
|
||||
\> ! transfer $v_{l-1}$\ to the next coarser level\\
|
||||
\> $v_l = R_lv_{l-1}$\ \\[1mm]
|
||||
\textbf{endfor} \\[2mm]
|
||||
! apply the coarsest-level correction\\[1mm]
|
||||
$y_{nlev} = A_{nlev}^{-1} v_{nlev}$\\[2mm]
|
||||
\textbf{for $l=nlev -1 , 1, -1$\ do}\\[1mm]
|
||||
\> ! transfer $y_{l+1}$\ to the next finer level\\
|
||||
\> $y_l = R_{l+1}^T y_{l+1}$;\\[1mm]
|
||||
\> ! compute the residual at the current level\\
|
||||
\> $r_l = v_l-A_l^{-1} y_l$;\\[1mm]
|
||||
\> ! apply the basic Schwarz preconditioner to the residual\\
|
||||
\> $r_l = M_l^{-1} r_l$\\[1mm]
|
||||
\> ! update $y_l$\\
|
||||
\> $y_l = y_l+r_l$\\
|
||||
\textbf{endfor} \\[1mm]
|
||||
$w = y_1$;
|
||||
\end{tabbing}
|
||||
}
|
||||
\end{minipage}
|
||||
}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img80.png"
|
||||
ALT="$S \in \Re^{n \times n}$">:
|
||||
<BR>
|
||||
<DIV ALIGN="RIGHT">
|
||||
WIDTH="430" HEIGHT="435" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img66.png"
|
||||
ALT="\framebox{
|
||||
\begin{minipage}{.85\textwidth} {\small
|
||||
\begin{tabbing}
|
||||
\quad \=\quad...
|
||||
...= y_l+r_l$\\
|
||||
\textbf{endfor} [1mm]
|
||||
$w = y_1$;
|
||||
\end{tabbing}}
|
||||
\end{minipage}}">
|
||||
|
||||
<!-- 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="img81.png"
|
||||
ALT="\begin{displaymath}
|
||||
P_C = S P,
|
||||
\end{displaymath}"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
(3)</TD></TR>
|
||||
</DIV></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="node24.html#BREZINA_VANEK">1</A>,<A
|
||||
HREF="node24.html#StubenGMD69_99">22</A>].
|
||||
A simple choice for <IMG
|
||||
WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img82.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="126" HEIGHT="30" BORDER="0"
|
||||
SRC="img83.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="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img84.png"
|
||||
ALT="$\omega$"> can be chosen
|
||||
using some estimate of the spectral radius of <IMG
|
||||
WIDTH="50" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img85.png"
|
||||
ALT="$D^{-1}A$"> [<A
|
||||
HREF="node24.html#BREZINA_VANEK">1</A>].
|
||||
</DIV>
|
||||
|
||||
<P>
|
||||
<HR>
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html204"
|
||||
<A NAME="tex2html209"
|
||||
HREF="node13.html">
|
||||
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
||||
SRC="file:/usr/share/latex2html/icons/next.png"></A>
|
||||
<A NAME="tex2html200"
|
||||
HREF="node10.html">
|
||||
<A NAME="tex2html205"
|
||||
HREF="node11.html">
|
||||
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
||||
SRC="file:/usr/share/latex2html/icons/up.png"></A>
|
||||
<A NAME="tex2html196"
|
||||
<A NAME="tex2html199"
|
||||
HREF="node11.html">
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||||
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
||||
SRC="file:/usr/share/latex2html/icons/prev.png"></A>
|
||||
<A NAME="tex2html202"
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<A NAME="tex2html207"
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||||
HREF="node2.html">
|
||||
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
||||
SRC="file:/usr/share/latex2html/icons/contents.png"></A>
|
||||
<BR>
|
||||
<B> Next:</B> <A NAME="tex2html205"
|
||||
HREF="node13.html">Getting Started</A>
|
||||
<B> Up:</B> <A NAME="tex2html201"
|
||||
HREF="node10.html">Multi-level Domain Decomposition Background</A>
|
||||
<B> Previous:</B> <A NAME="tex2html197"
|
||||
HREF="node11.html">Multi-level Schwarz Preconditioners</A>
|
||||
<B> <A NAME="tex2html203"
|
||||
<B> Next:</B> <A NAME="tex2html210"
|
||||
HREF="node13.html">Smoothed Aggregation</A>
|
||||
<B> Up:</B> <A NAME="tex2html206"
|
||||
HREF="node11.html">Multi-level Domain Decomposition Background</A>
|
||||
<B> Previous:</B> <A NAME="tex2html200"
|
||||
HREF="node11.html">Multi-level Domain Decomposition Background</A>
|
||||
<B> <A NAME="tex2html208"
|
||||
HREF="node2.html">Contents</A></B>
|
||||
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|
||||
|
||||
|
||||
Reference in New Issue
Block a user