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config/pac.m4 configure docs/html/WARNINGS docs/html/images.log docs/html/images.pl docs/html/images.tex docs/html/img100.png docs/html/img101.png docs/html/img92.png docs/html/img93.png docs/html/img94.png docs/html/img95.png docs/html/img96.png docs/html/img97.png docs/html/img98.png docs/html/img99.png docs/html/index.html docs/html/internals.pl docs/html/labels.pl docs/html/node1.html docs/html/node10.html docs/html/node11.html docs/html/node12.html docs/html/node13.html docs/html/node14.html docs/html/node15.html docs/html/node16.html docs/html/node17.html docs/html/node18.html docs/html/node19.html docs/html/node2.html docs/html/node20.html docs/html/node21.html docs/html/node22.html docs/html/node23.html docs/html/node24.html docs/html/node25.html docs/html/node26.html docs/html/node27.html docs/html/node28.html docs/html/node29.html docs/html/node3.html docs/html/node30.html docs/html/node31.html docs/html/node4.html docs/html/node5.html docs/html/node6.html docs/html/node7.html docs/html/node8.html docs/html/node9.html docs/html/userhtml.html docs/pdf/Makefile docs/pdf/abstract.tex docs/pdf/background.tex docs/pdf/bibliography.tex docs/pdf/building.tex docs/pdf/conventions.tex docs/pdf/distribution.tex docs/pdf/errors.tex docs/pdf/gettingstarted.tex docs/pdf/overview.tex docs/pdf/title.tex docs/pdf/userguide.tex docs/pdf/userhtml.tex docs/pdf/userinterface.tex Configure minro fix: require psblas 2.3 Doc fixes: dual version of title for pdf/html, fixed tables.
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -25,34 +25,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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</HEAD>
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<BODY >
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<H2><A NAME="SECTION00061000000000000000"></A><A NAME="sec:multilevel"></A>
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@@ -70,157 +71,157 @@ 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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<SPAN CLASS="MATH"><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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ALT="$A=(a_{ij}) \in \Re^{n \times n}$"></SPAN> is a
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nonsingular sparse matrix with a symmetric nonzero pattern,
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let <IMG
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let <SPAN CLASS="MATH"><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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ALT="$G=(W,E)$"></SPAN> be the adjacency graph of <SPAN CLASS="MATH"><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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ALT="$A$"></SPAN>, where <!-- MATH
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$W=\{1, 2, \ldots, n\}$
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-->
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<IMG
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<SPAN CLASS="MATH"><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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ALT="$W=\{1, 2, \ldots, n\}$"></SPAN>
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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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<SPAN CLASS="MATH"><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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ALT="$E=\{(i,j) : a_{ij} \neq 0\}$"></SPAN> are the vertex set and the edge set of <SPAN CLASS="MATH"><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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ALT="$G$"></SPAN>,
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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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them. For any integer <SPAN CLASS="MATH"><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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ALT="$\delta > 0$"></SPAN>, a <SPAN CLASS="MATH"><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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ALT="$\delta$"></SPAN>-overlap
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partition of <SPAN CLASS="MATH"><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="$W$"> can be defined recursively as follows.
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Given a 0-overlap (or non-overlapping) partition of <IMG
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ALT="$W$"></SPAN> can be defined recursively as follows.
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Given a 0-overlap (or non-overlapping) partition of <SPAN CLASS="MATH"><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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ALT="$W$"></SPAN>,
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i.e. a set of <SPAN CLASS="MATH"><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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ALT="$m$"></SPAN> 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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<SPAN CLASS="MATH"><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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ALT="$W_i^0 \subset W$"></SPAN> 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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<SPAN CLASS="MATH"><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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ALT="$\cup_{i=1}^m W_i^0 = W$"></SPAN>, a <SPAN CLASS="MATH"><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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ALT="$\delta$"></SPAN>-overlap
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partition of <SPAN CLASS="MATH"><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
|
||||
ALT="$W$"></SPAN> 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
|
||||
<SPAN CLASS="MATH"><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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ALT="$W_i^\delta \supset W_i^{\delta-1}$"></SPAN> 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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<SPAN CLASS="MATH"><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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ALT="$W_i^{\delta-1}$"></SPAN>.
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<P>
|
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Let <IMG
|
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Let <SPAN CLASS="MATH"><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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ALT="$n_i^\delta$"></SPAN> be the size of <SPAN CLASS="MATH"><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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ALT="$W_i^\delta$"></SPAN> 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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<SPAN CLASS="MATH"><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
|
||||
\Re^{n_i^\delta \times n}$"></SPAN> the restriction operator that maps
|
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a vector <SPAN CLASS="MATH"><IMG
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WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img19.png"
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ALT="$v \in \Re^n$"> onto the vector <!-- MATH
|
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ALT="$v \in \Re^n$"></SPAN> onto the vector <!-- MATH
|
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$v_i^{\delta} \in \Re^{n_i^\delta}$
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-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
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WIDTH="70" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img20.png"
|
||||
ALT="$v_i^{\delta} \in \Re^{n_i^\delta}$">
|
||||
containing the components of <IMG
|
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ALT="$v_i^{\delta} \in \Re^{n_i^\delta}$"></SPAN>
|
||||
containing the components of <SPAN CLASS="MATH"><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
|
||||
<IMG
|
||||
ALT="$v$"></SPAN> corresponding to the vertices in
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="30" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img17.png"
|
||||
ALT="$W_i^\delta$">. The transpose of <IMG
|
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ALT="$W_i^\delta$"></SPAN>. The transpose of <SPAN CLASS="MATH"><IMG
|
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WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img22.png"
|
||||
ALT="$R_i^{\delta}$"> is a
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ALT="$R_i^{\delta}$"></SPAN> is a
|
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prolongation operator from <!-- MATH
|
||||
$\Re^{n_i^\delta}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="32" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img23.png"
|
||||
ALT="$\Re^{n_i^\delta}$"> to <IMG
|
||||
ALT="$\Re^{n_i^\delta}$"></SPAN> to <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="26" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img24.png"
|
||||
ALT="$\Re^n$">.
|
||||
ALT="$\Re^n$"></SPAN>.
|
||||
The matrix <!-- MATH
|
||||
$A_i^\delta=R_i^\delta A (R_i^\delta)^T \in
|
||||
\Re^{n_i^\delta \times n_i^\delta}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="201" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img25.png"
|
||||
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
|
||||
\Re^{n_i^\delta \times n_i^\delta}$"></SPAN> can be considered
|
||||
as a restriction of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$A$"> corresponding to the set <IMG
|
||||
ALT="$A$"></SPAN> corresponding to the set <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img26.png"
|
||||
ALT="$W_i^{\delta}$">.
|
||||
ALT="$W_i^{\delta}$"></SPAN>.
|
||||
|
||||
<P>
|
||||
The <I>classical one-level AS</I> preconditioner is defined by
|
||||
The <SPAN CLASS="textit">classical one-level AS</SPAN> preconditioner is defined by
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
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@@ -238,105 +239,105 @@ M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
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||||
</DIV>
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<BR CLEAR="ALL">
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<P></P>
|
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where <IMG
|
||||
where <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img28.png"
|
||||
ALT="$A_i^\delta$"> is assumed to be nonsingular. Its application
|
||||
to a vector <IMG
|
||||
ALT="$A_i^\delta$"></SPAN> is assumed to be nonsingular. Its application
|
||||
to a vector <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img19.png"
|
||||
ALT="$v \in \Re^n$"> within a Krylov solver requires the following
|
||||
ALT="$v \in \Re^n$"></SPAN> within a Krylov solver requires the following
|
||||
three steps:
|
||||
|
||||
<OL>
|
||||
<LI>restriction of <IMG
|
||||
<LI>restriction of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$"> as <!-- MATH
|
||||
ALT="$v$"></SPAN> as <!-- MATH
|
||||
$v_i = R_i^{\delta} v$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img29.png"
|
||||
ALT="$v_i = R_i^{\delta} v$">, <IMG
|
||||
ALT="$v_i = R_i^{\delta} v$"></SPAN>, <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="97" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img30.png"
|
||||
ALT="$i=1,\ldots,m$">;
|
||||
ALT="$i=1,\ldots,m$"></SPAN>;
|
||||
</LI>
|
||||
<LI>solution of the linear systems <!-- MATH
|
||||
$A_i^\delta w_i = v_i$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="80" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img31.png"
|
||||
ALT="$A_i^\delta w_i = v_i$">,
|
||||
<IMG
|
||||
ALT="$A_i^\delta w_i = v_i$"></SPAN>,
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="97" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img30.png"
|
||||
ALT="$i=1,\ldots,m$">;
|
||||
ALT="$i=1,\ldots,m$"></SPAN>;
|
||||
</LI>
|
||||
<LI>prolongation and sum of the <IMG
|
||||
<LI>prolongation and sum of the <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img32.png"
|
||||
ALT="$w_i$">'s, i.e. <!-- MATH
|
||||
ALT="$w_i$"></SPAN>'s, i.e. <!-- MATH
|
||||
$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="144" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img33.png"
|
||||
ALT="$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$">.
|
||||
ALT="$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$"></SPAN>.
|
||||
</LI>
|
||||
</OL>
|
||||
Note that the linear systems at step 2 are usually solved approximately,
|
||||
e.g. using incomplete LU factorizations such as ILU(<IMG
|
||||
e.g. using incomplete LU factorizations such as ILU(<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img34.png"
|
||||
ALT="$p$">), MILU(<IMG
|
||||
ALT="$p$"></SPAN>), MILU(<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img34.png"
|
||||
ALT="$p$">) and
|
||||
ILU(<IMG
|
||||
ALT="$p$"></SPAN>) and
|
||||
ILU(<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img35.png"
|
||||
ALT="$p,t$">) [<A
|
||||
HREF="node30.html#Saad_book">14</A>, Chapter 10].
|
||||
ALT="$p,t$"></SPAN>) [<A
|
||||
HREF="node30.html#Saad_book">19</A>, Chapter 10].
|
||||
|
||||
<P>
|
||||
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
|
||||
time on parallel distributed-memory computers is the so-called <SPAN CLASS="textit">Restricted AS
|
||||
(RAS)</SPAN> preconditioner [<A
|
||||
HREF="node30.html#CAI_SARKIS">5</A>,<A
|
||||
HREF="node30.html#EFSTATHIOU">10</A>]. It
|
||||
is obtained by zeroing the components of <IMG
|
||||
HREF="node30.html#EFSTATHIOU">13</A>]. It
|
||||
is obtained by zeroing the components of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img32.png"
|
||||
ALT="$w_i$"> corresponding to the
|
||||
ALT="$w_i$"></SPAN> 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
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
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
|
||||
ALT="$(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$"></SPAN>,
|
||||
where <SPAN CLASS="MATH"><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
|
||||
ALT="$\tilde{R}_i^0$"></SPAN> is obtained by zeroing the rows of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img38.png"
|
||||
ALT="$R_i^\delta$">
|
||||
ALT="$R_i^\delta$"></SPAN>
|
||||
corresponding to the vertices in <!-- MATH
|
||||
$W_i^\delta \backslash W_i^0$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="66" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img39.png"
|
||||
ALT="$W_i^\delta \backslash W_i^0$">:
|
||||
ALT="$W_i^\delta \backslash W_i^0$"></SPAN>:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
|
||||
@@ -354,10 +355,10 @@ M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
Analogously, the AS variant called <I>AS with Harmonic extension (ASH)</I>
|
||||
Analogously, the AS variant called <SPAN CLASS="textit">AS with Harmonic extension (ASH)</SPAN>
|
||||
is defined by
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
||||
@@ -374,57 +375,57 @@ M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
We note that for <IMG
|
||||
We note that for <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="45" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img42.png"
|
||||
ALT="$\delta=0$"> the three variants of the AS preconditioner are
|
||||
ALT="$\delta=0$"></SPAN> 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
|
||||
preconditioned iterative solvers deteriorates as the number <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img11.png"
|
||||
ALT="$m$"> of partitions
|
||||
of <IMG
|
||||
ALT="$m$"></SPAN> of partitions
|
||||
of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img10.png"
|
||||
ALT="$W$"> increases [<A
|
||||
ALT="$W$"></SPAN> increases [<A
|
||||
HREF="node30.html#dd1_94">7</A>,<A
|
||||
HREF="node30.html#dd2_96">15</A>]. To reduce the dependency
|
||||
HREF="node30.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
|
||||
a coarse-space approximation <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="$A_C$"> of the matrix <IMG
|
||||
ALT="$A_C$"></SPAN> of the matrix <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$A$">.
|
||||
In a pure algebraic setting, <IMG
|
||||
ALT="$A$"></SPAN>.
|
||||
In a pure algebraic setting, <SPAN CLASS="MATH"><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
|
||||
ALT="$A_C$"></SPAN> is usually built with
|
||||
a Galerkin approach. Given a set <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img44.png"
|
||||
ALT="$W_C$"> of <I>coarse vertices</I>,
|
||||
with size <IMG
|
||||
ALT="$W_C$"></SPAN> of <SPAN CLASS="textit">coarse vertices</SPAN>,
|
||||
with size <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="26" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img45.png"
|
||||
ALT="$n_C$">, and a suitable restriction operator
|
||||
ALT="$n_C$"></SPAN>, and a suitable restriction operator
|
||||
<!-- MATH
|
||||
$R_C \in \Re^{n_C \times n}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="101" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img46.png"
|
||||
ALT="$R_C \in \Re^{n_C \times n}$">, <IMG
|
||||
ALT="$R_C \in \Re^{n_C \times n}$"></SPAN>, <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="$A_C$"> is defined as
|
||||
ALT="$A_C$"></SPAN> is defined as
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
A_C=R_C A R_C^T
|
||||
@@ -441,12 +442,12 @@ A_C=R_C A R_C^T
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
and the coarse-level correction matrix to be combined with a generic
|
||||
one-level AS preconditioner <IMG
|
||||
one-level AS preconditioner <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img48.png"
|
||||
ALT="$M_{1L}$"> is obtained as
|
||||
ALT="$M_{1L}$"></SPAN> is obtained as
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{C}^{-1}= R_C^T A_C^{-1} R_C,
|
||||
@@ -462,36 +463,36 @@ M_{C}^{-1}= R_C^T A_C^{-1} R_C,
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
where <IMG
|
||||
where <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="$A_C$"> is assumed to be nonsingular. The application of <IMG
|
||||
ALT="$A_C$"></SPAN> is assumed to be nonsingular. The application of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img50.png"
|
||||
ALT="$M_{C}^{-1}$">
|
||||
to a vector <IMG
|
||||
ALT="$M_{C}^{-1}$"></SPAN>
|
||||
to a vector <SPAN CLASS="MATH"><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
|
||||
ALT="$v$"></SPAN> corresponds to a restriction, a solution and
|
||||
a prolongation step; the solution step, involving the matrix <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="$A_C$">,
|
||||
ALT="$A_C$"></SPAN>,
|
||||
may be carried out also approximately.
|
||||
|
||||
<P>
|
||||
The combination of <IMG
|
||||
The combination of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img51.png"
|
||||
ALT="$M_{C}$"> and <IMG
|
||||
ALT="$M_{C}$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img48.png"
|
||||
ALT="$M_{1L}$"> may be
|
||||
ALT="$M_{1L}$"></SPAN> may be
|
||||
performed in either an additive or a multiplicative framework.
|
||||
In the former case, the <I>two-level additive</I> Schwarz preconditioner
|
||||
In the former case, the <SPAN CLASS="textit">two-level additive</SPAN> Schwarz preconditioner
|
||||
is obtained:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
|
||||
@@ -507,38 +508,38 @@ M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
Applying <IMG
|
||||
Applying <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="59" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img53.png"
|
||||
ALT="$M_{2L-A}^{-1}$"> to a vector <IMG
|
||||
ALT="$M_{2L-A}^{-1}$"></SPAN> to a vector <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$"> within a Krylov solver
|
||||
corresponds to applying <IMG
|
||||
ALT="$v$"></SPAN> within a Krylov solver
|
||||
corresponds to applying <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img50.png"
|
||||
ALT="$M_{C}^{-1}$">
|
||||
and <IMG
|
||||
ALT="$M_{C}^{-1}$"></SPAN>
|
||||
and <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img54.png"
|
||||
ALT="$M_{1L}^{-1}$"> to <IMG
|
||||
ALT="$M_{1L}^{-1}$"></SPAN> to <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$"> independently and then summing up
|
||||
ALT="$v$"></SPAN> independently and then summing up
|
||||
the results.
|
||||
|
||||
<P>
|
||||
In the multiplicative case, the combination can be
|
||||
performed by first applying the smoother <IMG
|
||||
performed by first applying the smoother <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img54.png"
|
||||
ALT="$M_{1L}^{-1}$"> and then
|
||||
the coarse-level correction operator <IMG
|
||||
ALT="$M_{1L}^{-1}$"></SPAN> and then
|
||||
the coarse-level correction operator <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img50.png"
|
||||
ALT="$M_{C}^{-1}$">:
|
||||
ALT="$M_{C}^{-1}$"></SPAN>:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
\begin{array}{l}
|
||||
@@ -559,10 +560,10 @@ z = w + M_{C}^{-1} (v-Aw);
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
this corresponds to the following <I>two-level hybrid pre-smoothed</I>
|
||||
this corresponds to the following <SPAN CLASS="textit">two-level hybrid pre-smoothed</SPAN>
|
||||
Schwarz preconditioner:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
|
||||
@@ -581,7 +582,7 @@ M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
|
||||
On the other hand, by applying the smoother after the coarse-level correction,
|
||||
i.e. by computing
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
\begin{array}{l}
|
||||
@@ -602,10 +603,10 @@ z = w + M_{1L}^{-1} (v-Aw) ,
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
the <I>two-level hybrid post-smoothed</I>
|
||||
the <SPAN CLASS="textit">two-level hybrid post-smoothed</SPAN>
|
||||
Schwarz preconditioner is obtained:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
|
||||
@@ -623,16 +624,16 @@ M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
|
||||
<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
|
||||
preconditioner is symmetric if <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$A$">, <IMG
|
||||
ALT="$A$"></SPAN>, <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img48.png"
|
||||
ALT="$M_{1L}$"> and <IMG
|
||||
ALT="$M_{1L}$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img51.png"
|
||||
ALT="$M_{C}$"> are symmetric.
|
||||
ALT="$M_{C}$"></SPAN> are symmetric.
|
||||
|
||||
<P>
|
||||
As previously noted, on parallel computers the number of submatrices usually matches
|
||||
@@ -643,52 +644,51 @@ On the other hand, the use of few processors often leads to local sumatrices tha
|
||||
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>
|
||||
coarse-level system. The corresponding preconditioners, called <SPAN CLASS="textit">multi-level</SPAN>
|
||||
preconditioners, can significantly reduce the computational cost of preconditioning
|
||||
with respect to the two-level case (see [<A
|
||||
HREF="node30.html#dd2_96">15</A>, Chapter 3]).
|
||||
HREF="node30.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="node30.html#dd2_96">15</A>, Chapter 3].
|
||||
HREF="node30.html#dd2_96">20</A>, Chapter 3].
|
||||
The algorithm for the application of a multi-level hybrid
|
||||
post-smoothed preconditioner <IMG
|
||||
post-smoothed preconditioner <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img59.png"
|
||||
ALT="$M$"> to a vector <IMG
|
||||
ALT="$M$"></SPAN> to a vector <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$">, i.e. for the
|
||||
computation of <IMG
|
||||
ALT="$v$"></SPAN>, i.e. for the
|
||||
computation of <SPAN CLASS="MATH"><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"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>. Here the number of levels
|
||||
is denoted by <IMG
|
||||
ALT="$w=M^{-1}v$"></SPAN>, is reported, for
|
||||
example, in Figure <A HREF="#fig:mlhpost_alg">1</A>. Here the number of levels
|
||||
is denoted by <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="37" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img61.png"
|
||||
ALT="$nlev$"> and the levels are numbered in increasing order starting
|
||||
ALT="$nlev$"></SPAN> 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
|
||||
and the corresponding basic preconditioner at each level <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="10" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img62.png"
|
||||
ALT="$l$"> are denoted by <IMG
|
||||
ALT="$l$"></SPAN> are denoted by <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="22" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img63.png"
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ALT="$A_l$"> and
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<IMG
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ALT="$A_l$"></SPAN> and
|
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<SPAN CLASS="MATH"><IMG
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WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img64.png"
|
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ALT="$M_l$">, respectively, with <IMG
|
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ALT="$M_l$"></SPAN>, respectively, with <SPAN CLASS="MATH"><IMG
|
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WIDTH="61" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img65.png"
|
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ALT="$A_1=A$">.
|
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ALT="$A_1=A$"></SPAN>.
|
||||
|
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<DIV ALIGN="CENTER"><A NAME="fig:mlhpost_alg"></A><A NAME="557"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:mlhpost_alg"></A><A NAME="508"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure:</STRONG>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
|
||||
Application of the multi-level hybrid post-smoothed preconditioner.</CAPTION>
|
||||
<TR><TD>
|
||||
<DIV ALIGN="CENTER">
|
||||
@@ -728,7 +728,7 @@ $w = y_1$;
|
||||
\begin{tabbing}
|
||||
\quad \=\quad...
|
||||
...= y_l+r_l$\\
|
||||
\textbf{endfor} \ [1mm]
|
||||
\textbf{endfor} [1mm]
|
||||
$w = y_1$;
|
||||
\end{tabbing}}
|
||||
\end{minipage}}">
|
||||
@@ -738,37 +738,38 @@ $w = y_1$;
|
||||
</DIV>
|
||||
|
||||
<P>
|
||||
<HR>
|
||||
|
||||
<DIV CLASS="navigation"><HR>
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html201"
|
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<A NAME="tex2html200"
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HREF="node12.html">
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<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
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SRC="file:/usr/share/latex2html/icons/next.png"></A>
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<A NAME="tex2html197"
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<A NAME="tex2html196"
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HREF="node10.html">
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<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
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SRC="file:/usr/share/latex2html/icons/up.png"></A>
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<A NAME="tex2html191"
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<A NAME="tex2html190"
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HREF="node10.html">
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<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
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SRC="file:/usr/share/latex2html/icons/prev.png"></A>
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<A NAME="tex2html199"
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HREF="node1.html">
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<A NAME="tex2html198"
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HREF="node2.html">
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<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
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SRC="file:/usr/share/latex2html/icons/contents.png"></A>
|
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<BR>
|
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<B> Next:</B> <A NAME="tex2html202"
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<B> Next:</B> <A NAME="tex2html201"
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HREF="node12.html">Smoothed Aggregation</A>
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<B> Up:</B> <A NAME="tex2html198"
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<B> Up:</B> <A NAME="tex2html197"
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HREF="node10.html">Multi-level Domain Decomposition Background</A>
|
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<B> Previous:</B> <A NAME="tex2html192"
|
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<B> Previous:</B> <A NAME="tex2html191"
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HREF="node10.html">Multi-level Domain Decomposition Background</A>
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||||
<B> <A NAME="tex2html200"
|
||||
HREF="node1.html">Contents</A></B>
|
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<B> <A NAME="tex2html199"
|
||||
HREF="node2.html">Contents</A></B> </DIV>
|
||||
<!--End of Navigation Panel-->
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
||||
</BODY>
|
||||
</HTML>
|
||||
|
||||
Reference in New Issue
Block a user