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mld2p4:
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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Salvatore Filippone
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@@ -1,4 +1,4 @@
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
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original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -24,34 +24,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<BR>
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<B> Next:</B> <A NAME="tex2html212"
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<B> Next:</B> <A NAME="tex2html211"
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HREF="node13.html">Getting Started</A>
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<B> Up:</B> <A NAME="tex2html208"
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<B> Up:</B> <A NAME="tex2html207"
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HREF="node10.html">Multi-level Domain Decomposition Background</A>
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<B> Previous:</B> <A NAME="tex2html204"
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<B> Previous:</B> <A NAME="tex2html203"
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HREF="node11.html">Multi-level Schwarz Preconditioners</A>
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<B> <A NAME="tex2html210"
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HREF="node1.html">Contents</A></B>
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<BR>
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<B> <A NAME="tex2html209"
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HREF="node2.html">Contents</A></B>
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<BR>
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<BR></DIV>
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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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@@ -60,29 +61,29 @@ Smoothed Aggregation
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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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In order to define the restriction operator <SPAN CLASS="MATH"><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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ALT="$R_C$"></SPAN>, which is used to compute
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the coarse-level matrix <SPAN CLASS="MATH"><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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ALT="$A_C$"></SPAN>, MLD2P4 uses the <SPAN CLASS="textit">smoothed aggregation</SPAN>
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algorithm described in [<A
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HREF="node30.html#BREZINA_VANEK">1</A>,<A
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HREF="node30.html#VANEK_MANDEL_BREZINA">19</A>].
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HREF="node30.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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<IMG
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<SPAN CLASS="MATH"><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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ALT="$W_C$"></SPAN> by suitably grouping the vertices 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="$W$"> into disjoint subsets
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(aggregates), and to define the coarse-to-fine space transfer operator <IMG
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ALT="$W$"></SPAN> into disjoint subsets
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(aggregates), and to define the coarse-to-fine space transfer operator <SPAN CLASS="MATH"><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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ALT="$R_C^T$"></SPAN> 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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@@ -90,29 +91,29 @@ prolongation operator, to improve the quality of the coarse-space correction.
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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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<LI>coarsening of the vertex set <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="$W$">, to obtain <IMG
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ALT="$W$"></SPAN>, to obtain <SPAN CLASS="MATH"><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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ALT="$W_C$"></SPAN>;
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</LI>
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<LI>construction of the prolongator <IMG
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<LI>construction of the prolongator <SPAN CLASS="MATH"><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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ALT="$R_C^T$"></SPAN>;
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</LI>
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<LI>application of <IMG
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<LI>application of <SPAN CLASS="MATH"><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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ALT="$R_C$"></SPAN> and <SPAN CLASS="MATH"><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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ALT="$R_C^T$"></SPAN> to build <SPAN CLASS="MATH"><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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ALT="$A_C$"></SPAN>.
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</LI>
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</OL>
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@@ -120,20 +121,20 @@ Three main steps can be identified in the smoothed aggregation procedure:
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To perform the coarsening step, we have implemented the aggregation algorithm sketched
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in [<A
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HREF="node30.html#apnum_07">4</A>]. According to [<A
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HREF="node30.html#VANEK_MANDEL_BREZINA">19</A>], a modification of
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HREF="node30.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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in which each aggregate <SPAN CLASS="MATH"><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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ALT="$N_r$"></SPAN> is made of vertices 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="$W$"> that are <I>strongly coupled</I>
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to a certain root vertex <IMG
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ALT="$W$"></SPAN> that are <SPAN CLASS="textit">strongly coupled</SPAN>
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to a certain root vertex <SPAN CLASS="MATH"><IMG
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WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img70.png"
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ALT="$r \in W$">, i.e. <BR><P></P>
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<DIV ALIGN="CENTER">
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ALT="$r \in W$"></SPAN>, i.e. <BR><P></P>
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<DIV ALIGN="CENTER" CLASS="mathdisplay">
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<!-- MATH
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\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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@@ -153,47 +154,47 @@ N_r = \left\{s \in W: |a_{rs}| > \theta \sqrt{|a_{rr}a_{ss}|} \right\}
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for a given <!-- MATH
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$\theta \in [0,1]$
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-->
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<IMG
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<SPAN CLASS="MATH"><IMG
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WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img72.png"
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ALT="$\theta \in [0,1]$">.
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Since this algorithm has a sequential nature, a <I>decoupled</I> version of
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it has been chosen, where each processor <IMG
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ALT="$\theta \in [0,1]$"></SPAN>.
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Since this algorithm has a sequential nature, a <SPAN CLASS="textit">decoupled</SPAN> version of
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it has been chosen, where each processor <SPAN CLASS="MATH"><IMG
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WIDTH="10" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img73.png"
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ALT="$i$"> independently applies the algorithm to
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the set of vertices <IMG
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ALT="$i$"></SPAN> independently applies the algorithm to
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the set of vertices <SPAN CLASS="MATH"><IMG
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WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img74.png"
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ALT="$W_i^0$"> assigned to it in the initial data distribution. This
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ALT="$W_i^0$"></SPAN> assigned to it in the initial data distribution. This
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version is embarrassingly parallel, since it does not require any data communication.
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On the other hand, it may produce non-uniform aggregates near boundary vertices,
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i.e. near vertices adjacent to vertices in other processors, and is strongly
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dependent on the number of processors and on the initial partitioning of the matrix <IMG
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dependent on the number of processors and on the initial partitioning of the matrix <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="$A$">.
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ALT="$A$"></SPAN>.
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Nevertheless, this algorithm has been chosen for the implementation in MLD2P4,
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since it has been shown to produce good results in practice
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[<A
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HREF="node30.html#aaecc_07">3</A>,<A
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HREF="node30.html#apnum_07">4</A>,<A
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HREF="node30.html#TUMINARO_TONG">18</A>].
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HREF="node30.html#TUMINARO_TONG">23</A>].
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<P>
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The prolongator <IMG
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The prolongator <SPAN CLASS="MATH"><IMG
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WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
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SRC="img75.png"
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ALT="$P_C=R_C^T$"> is built starting from a <I>tentative prolongator</I>
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ALT="$P_C=R_C^T$"></SPAN> is built starting from a <SPAN CLASS="textit">tentative prolongator</SPAN>
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<!-- MATH
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$P \in \Re^{n \times n_C}$
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-->
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<IMG
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<SPAN CLASS="MATH"><IMG
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WIDTH="90" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
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SRC="img76.png"
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ALT="$P \in \Re^{n \times n_C}$">, defined as
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ALT="$P \in \Re^{n \times n_C}$"></SPAN>, defined as
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<BR>
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<DIV ALIGN="RIGHT">
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<DIV ALIGN="RIGHT" CLASS="mathdisplay">
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<!-- MATH
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\begin{equation}
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@@ -204,6 +205,7 @@ P=(p_{ij}), \quad p_{ij}=
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\end{array} \right. .
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\end{equation}
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-->
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<A NAME="eq:tent_prol"></A>
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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="290" HEIGHT="52" BORDER="0"
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@@ -216,32 +218,33 @@ P=(p_{ij}), \quad p_{ij}=
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0 & \quad \mbox{otherwise}
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\end{array} \right. .
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\end{displaymath}"></TD>
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<TD WIDTH=10 ALIGN="RIGHT">
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(2)</TD></TR>
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<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
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(<SPAN CLASS="arabic">2</SPAN>)</TD></TR>
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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P>
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<IMG
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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="img78.png"
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ALT="$P_C$"> is obtained by
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applying to <IMG
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ALT="$P_C$"></SPAN> is obtained by
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applying to <SPAN CLASS="MATH"><IMG
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img79.png"
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ALT="$P$"> a smoother <!-- MATH
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ALT="$P$"></SPAN> a smoother <!-- MATH
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$S \in \Re^{n \times n}$
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-->
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<IMG
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<SPAN CLASS="MATH"><IMG
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WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$S \in \Re^{n \times n}$">:
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ALT="$S \in \Re^{n \times n}$"></SPAN>:
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<BR>
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<DIV ALIGN="RIGHT">
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<DIV ALIGN="RIGHT" CLASS="mathdisplay">
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<!-- MATH
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\begin{equation}
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P_C = S P,
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\end{equation}
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-->
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<A NAME="eq:smoothed_prol"></A>
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<TABLE WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:smoothed_prol"></A><IMG
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WIDTH="73" HEIGHT="30" BORDER="0"
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@@ -249,27 +252,28 @@ P_C = S P,
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ALT="\begin{displaymath}
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P_C = S P,
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\end{displaymath}"></TD>
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<TD WIDTH=10 ALIGN="RIGHT">
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(3)</TD></TR>
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<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
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(<SPAN CLASS="arabic">3</SPAN>)</TD></TR>
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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P>
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in order to remove oscillatory components from the range of the prolongator
|
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and hence to improve the convergence properties of the multi-level
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Schwarz method [<A
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||||
HREF="node30.html#BREZINA_VANEK">1</A>,<A
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HREF="node30.html#StubenGMD69_99">17</A>].
|
||||
A simple choice for <IMG
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||||
HREF="node30.html#StubenGMD69_99">22</A>].
|
||||
A simple choice for <SPAN CLASS="MATH"><IMG
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||||
WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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||||
SRC="img82.png"
|
||||
ALT="$S$"> is the damped Jacobi smoother:
|
||||
ALT="$S$"></SPAN> is the damped Jacobi smoother:
|
||||
<BR>
|
||||
<DIV ALIGN="RIGHT">
|
||||
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
|
||||
|
||||
<!-- MATH
|
||||
\begin{equation}
|
||||
S = I - \omega D^{-1} A ,
|
||||
\end{equation}
|
||||
-->
|
||||
<A NAME="eq:jac_smoother"></A>
|
||||
<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"
|
||||
@@ -277,52 +281,53 @@ S = I - \omega D^{-1} A ,
|
||||
ALT="\begin{displaymath}
|
||||
S = I - \omega D^{-1} A ,
|
||||
\end{displaymath}"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
(4)</TD></TR>
|
||||
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
||||
(<SPAN CLASS="arabic">4</SPAN>)</TD></TR>
|
||||
</TABLE>
|
||||
<BR CLEAR="ALL"></DIV><P></P>
|
||||
where the value of <IMG
|
||||
where the value of <SPAN CLASS="MATH"><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
|
||||
ALT="$\omega$"></SPAN> can be chosen
|
||||
using some estimate of the spectral radius of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="50" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img85.png"
|
||||
ALT="$D^{-1}A$"> [<A
|
||||
ALT="$D^{-1}A$"></SPAN> [<A
|
||||
HREF="node30.html#BREZINA_VANEK">1</A>].
|
||||
|
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<P>
|
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<HR>
|
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|
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<DIV CLASS="navigation"><HR>
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<!--Navigation Panel-->
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<A NAME="tex2html211"
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<A NAME="tex2html210"
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HREF="node13.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="tex2html207"
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<A NAME="tex2html206"
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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="tex2html203"
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<A NAME="tex2html202"
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HREF="node11.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="tex2html209"
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HREF="node1.html">
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<A NAME="tex2html208"
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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="tex2html212"
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<B> Next:</B> <A NAME="tex2html211"
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HREF="node13.html">Getting Started</A>
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<B> Up:</B> <A NAME="tex2html208"
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<B> Up:</B> <A NAME="tex2html207"
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HREF="node10.html">Multi-level Domain Decomposition Background</A>
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<B> Previous:</B> <A NAME="tex2html204"
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<B> Previous:</B> <A NAME="tex2html203"
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HREF="node11.html">Multi-level Schwarz Preconditioners</A>
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<B> <A NAME="tex2html210"
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HREF="node1.html">Contents</A></B>
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<B> <A NAME="tex2html209"
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HREF="node2.html">Contents</A></B> </DIV>
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<!--End of Navigation Panel-->
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<ADDRESS>
|
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Salvatore Filippone
|
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2008-07-22
|
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2008-07-23
|
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</ADDRESS>
|
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</BODY>
|
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</HTML>
|
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|
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Reference in New Issue
Block a user