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Fixed doc copyright vs contribution
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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>AMG preconditioners</TITLE>
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<META NAME="description" CONTENT="AMG 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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@@ -20,7 +20,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<LINK REL="next" HREF="node14.html">
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<LINK REL="previous" HREF="node12.html">
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<LINK REL="up" HREF="node11.html">
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<LINK REL="up" HREF="node12.html">
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<LINK REL="next" HREF="node14.html">
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</HEAD>
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@@ -30,7 +30,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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HREF="node14.html">
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<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A>
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<A NAME="tex2html237"
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HREF="node11.html">
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HREF="node12.html">
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<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A>
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<A NAME="tex2html231"
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HREF="node12.html">
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@@ -40,336 +40,247 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
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<BR>
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<B> Next:</B> <A NAME="tex2html242"
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HREF="node14.html">Smoothers and coarsest-level solvers</A>
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HREF="node14.html">Smoothed Aggregation</A>
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<B> Up:</B> <A NAME="tex2html238"
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HREF="node11.html">Multigrid Background</A>
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HREF="node12.html">Multigrid Background</A>
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<B> Previous:</B> <A NAME="tex2html232"
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HREF="node12.html">AMG preconditioners</A>
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HREF="node12.html">Multigrid Background</A>
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<B> <A NAME="tex2html240"
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HREF="node2.html">Contents</A></B>
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<BR>
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<BR>
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<!--End of Navigation Panel-->
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<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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AMG preconditioners
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</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to define the prolongator <IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$">, used to compute
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the coarse-level matrix <IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img15.png"
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ALT="$A^{k+1}$">, MLD2P4 uses the smoothed aggregation
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algorithm described in [<A
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HREF="node29.html#BREZINA_VANEK">2</A>,<A
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HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>].
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The basic idea of this algorithm is to build a coarse set of indices
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<IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img26.png"
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ALT="$\Omega^{k+1}$"> by suitably grouping the indices of <IMG
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WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\Omega^k$"> into disjoint
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subsets (aggregates), and to define the coarse-to-fine space transfer operator
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<IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$"> by applying a suitable smoother to a simple piecewise constant
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prolongation operator, with the aim of improving the quality of the coarse-space correction.
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to describe the AMG preconditioners available in MLD2P4, we consider a
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linear system
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</FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Three main steps can be identified in the smoothed aggregation procedure:
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</FONT></FONT></FONT>
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<OL>
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<LI>aggregation of the indices of <IMG
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WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\Omega^k$"> to obtain <IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img26.png"
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ALT="$\Omega^{k+1}$">;
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</LI>
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<LI>construction of the prolongator <IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$">;
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</LI>
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<LI>application of <IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$"> and <IMG
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WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img17.png"
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ALT="$R^k=(P^k)^T$"> to build <IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img15.png"
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ALT="$A^{k+1}$">.
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</LI>
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</OL><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to perform the coarsening step, the smoothed aggregation algorithm
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described in [<A
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HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>] is used. In this algorithm,
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each index <!-- MATH
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$j \in \Omega^{k+1}$
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<BR>
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<DIV ALIGN="RIGHT">
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<!-- MATH
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\begin{equation}
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Ax=b,
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\end{equation}
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-->
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<TABLE WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:system"></A><IMG
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WIDTH="58" HEIGHT="30" BORDER="0"
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SRC="img2.png"
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ALT="\begin{displaymath}
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Ax=b,
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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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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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where <!-- MATH
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$A=(a_{ij}) \in \mathbb{R}^{n \times n}$
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-->
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<IMG
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WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img27.png"
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ALT="$j \in \Omega^{k+1}$"> corresponds to an aggregate <IMG
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WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img28.png"
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ALT="$\Omega^k_j$"> of <IMG
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WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.png"
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ALT="$A=(a_{ij}) \in \mathbb{R}^{n \times n}$"> is a nonsingular sparse matrix;
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for ease of presentation we assume <IMG
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$A$"> is real, but the
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results are valid for the complex case as well.
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</FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Let us assume as finest index space the set of row (column) indices of <IMG
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$A$">, i.e.,
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<!-- MATH
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$\Omega = \{1, 2, \ldots, n\}$
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-->
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<IMG
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WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img6.png"
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ALT="$\Omega = \{1, 2, \ldots, n\}$">.
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Any algebraic multilevel preconditioners implemented in MLD2P4 generates
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a hierarchy of index spaces and a corresponding hierarchy of matrices,
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</FONT></FONT></FONT>
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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\Omega^1 \equiv \Omega \supset \Omega^2 \supset \ldots \supset \Omega^{nlev},
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\quad A^1 \equiv A, A^2, \ldots, A^{nlev},
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\end{displaymath}
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-->
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<IMG
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WIDTH="398" HEIGHT="30" BORDER="0"
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SRC="img7.png"
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ALT="\begin{displaymath}\Omega^1 \equiv \Omega \supset \Omega^2 \supset \ldots \supset \Omega^{nlev},␍\quad A^1 \equiv A, A^2, \ldots, A^{nlev}, \end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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by using the information contained in <IMG
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$A$">, without assuming any
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knowledge of the geometry of the problem from which <IMG
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$A$"> originates.
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A vector space <!-- MATH
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$\mathbb{R}^{n_{k}}$
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-->
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<IMG
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WIDTH="33" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img8.png"
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ALT="$\mathbb{R}^{n_{k}}$"> is associated with <IMG
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WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\Omega^k$">,
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consisting of a suitably chosen index <!-- MATH
|
||||
$i \in \Omega^k$
|
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-->
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<IMG
|
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WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img29.png"
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ALT="$i \in \Omega^k$"> and indices that are (usually) contained in a
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strongly-coupled neighborood of <IMG
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WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img30.png"
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ALT="$i$">, i.e.,
|
||||
</FONT></FONT></FONT>
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<BR>
|
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<DIV ALIGN="RIGHT">
|
||||
|
||||
<!-- MATH
|
||||
\begin{equation}
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\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
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\left\{ r \in \Omega^k: |a_{ir}^k| > \theta \sqrt{|a_{ii}^ka_{rr}^k|} \right \} \cup \left\{ i \right\},
|
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\end{equation}
|
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-->
|
||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:strongly_coup"></A><IMG
|
||||
WIDTH="387" HEIGHT="72" BORDER="0"
|
||||
SRC="img31.png"
|
||||
ALT="\begin{displaymath}
|
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\Omega^k_j \subset \mathcal{N}_i^k(\theta) = ␍ \left\{ r ...
|
||||
...vert a_{ii}^ka_{rr}^k\vert} \right \} \cup \left\{ i \right\},␍\end{displaymath}"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
(3)</TD></TR>
|
||||
</TABLE>
|
||||
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
|
||||
for a given threshold <!-- MATH
|
||||
$\theta \in [0,1]$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img32.png"
|
||||
ALT="$\theta \in [0,1]$"> (see [<A
|
||||
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>] for the details).
|
||||
Since this algorithm has a sequential nature, a decoupled
|
||||
version of it is applied, where each processor independently executes
|
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the algorithm on the set of indices assigned to it in the initial data
|
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distribution. This version is embarrassingly parallel, since it does not require any data
|
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communication. On the other hand, it may produce some nonuniform aggregates
|
||||
and is strongly dependent on the number of processors and on the initial partitioning
|
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of the matrix <IMG
|
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img3.png"
|
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ALT="$A$">. Nevertheless, this parallel algorithm has been chosen for
|
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MLD2P4, since it has been shown to produce good results in practice
|
||||
[<A
|
||||
HREF="node29.html#aaecc_07">5</A>,<A
|
||||
HREF="node29.html#apnum_07">7</A>,<A
|
||||
HREF="node29.html#TUMINARO_TONG">24</A>].
|
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</FONT></FONT></FONT>
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<P>
|
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The prolongator <IMG
|
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
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SRC="img25.png"
|
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ALT="$P^k$"> is built starting from a tentative prolongator
|
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<!-- MATH
|
||||
$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$
|
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-->
|
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<IMG
|
||||
WIDTH="117" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img33.png"
|
||||
ALT="$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$">, defined as
|
||||
</FONT></FONT></FONT>
|
||||
<BR>
|
||||
<DIV ALIGN="RIGHT">
|
||||
|
||||
<!-- MATH
|
||||
\begin{equation}
|
||||
\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
|
||||
\left\{ \begin{array}{ll}
|
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1 & \quad \mbox{if} \; i \in \Omega^k_j, \\
|
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0 & \quad \mbox{otherwise},
|
||||
\end{array} \right.
|
||||
\end{equation}
|
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-->
|
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<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:tent_prol"></A><IMG
|
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WIDTH="287" HEIGHT="51" BORDER="0"
|
||||
SRC="img34.png"
|
||||
ALT="\begin{displaymath}␍\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k = ␍\left\{...
|
||||
...ega^k_j, \ ␍0 & \quad \mbox{otherwise},␍\end{array} \right.
|
||||
\end{displaymath}"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
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(4)</TD></TR>
|
||||
</TABLE>
|
||||
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
|
||||
where <IMG
|
||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img28.png"
|
||||
ALT="$\Omega^k_j$"> is the aggregate of <IMG
|
||||
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img10.png"
|
||||
ALT="$n_k$"> is the size of <IMG
|
||||
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img9.png"
|
||||
ALT="$\Omega^k$">
|
||||
corresponding to the index <!-- MATH
|
||||
$j \in \Omega^{k+1}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img27.png"
|
||||
ALT="$j \in \Omega^{k+1}$">.
|
||||
<IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img25.png"
|
||||
ALT="$P^k$"> is obtained by applying to <IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img35.png"
|
||||
ALT="$\bar{P}^k$"> a smoother
|
||||
<!-- MATH
|
||||
$S^k \in \mathbb{R}^{n_k \times n_k}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img36.png"
|
||||
ALT="$S^k \in \mathbb{R}^{n_k \times n_k}$">:
|
||||
ALT="$\Omega^k$">.
|
||||
For all <IMG
|
||||
WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img11.png"
|
||||
ALT="$k < nlev$">, a restriction operator and a prolongation one are built,
|
||||
which connect two levels <IMG
|
||||
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img12.png"
|
||||
ALT="$k$"> and <IMG
|
||||
WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img13.png"
|
||||
ALT="$k+1$">:
|
||||
</FONT></FONT></FONT>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
P^k = S^k \bar{P}^k,
|
||||
P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad
|
||||
R^k \in \mathbb{R}^{n_{k+1}\times n_k};
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="90" HEIGHT="30" BORDER="0"
|
||||
SRC="img37.png"
|
||||
ALT="\begin{displaymath}␍P^k = S^k \bar{P}^k,␍\end{displaymath}">
|
||||
WIDTH="254" HEIGHT="30" BORDER="0"
|
||||
SRC="img14.png"
|
||||
ALT="\begin{displaymath}␍ P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad ␍ R^k \in \mathbb{R}^{n_{k+1}\times n_k};␍\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
|
||||
in order to remove nonsmooth components from the range of the prolongator,
|
||||
and hence to improve the convergence properties of the multi-level
|
||||
method [<A
|
||||
HREF="node29.html#BREZINA_VANEK">2</A>,<A
|
||||
HREF="node29.html#Stuben_01">23</A>].
|
||||
A simple choice for <IMG
|
||||
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img38.png"
|
||||
ALT="$S^k$"> is the damped Jacobi smoother:
|
||||
the matrix <IMG
|
||||
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img15.png"
|
||||
ALT="$A^{k+1}$"> is computed by using the previous operators according
|
||||
to the Galerkin approach, i.e.,
|
||||
</FONT></FONT></FONT>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
S^k = I - \omega^k (D^k)^{-1} A^k_F ,
|
||||
A^{k+1}=R^kA^kP^k.
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="175" HEIGHT="31" BORDER="0"
|
||||
SRC="img39.png"
|
||||
ALT="\begin{displaymath}␍S^k = I - \omega^k (D^k)^{-1} A^k_F , ␍\end{displaymath}">
|
||||
WIDTH="131" HEIGHT="27" BORDER="0"
|
||||
SRC="img16.png"
|
||||
ALT="\begin{displaymath}␍ A^{k+1}=R^kA^kP^k.␍\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
|
||||
where <IMG
|
||||
WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img40.png"
|
||||
ALT="$D^k$"> is the diagonal matrix with the same diagonal entries as <IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img41.png"
|
||||
ALT="$A^k$">,
|
||||
In the current implementation of MLD2P4 we have <IMG
|
||||
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img17.png"
|
||||
ALT="$R^k=(P^k)^T$">
|
||||
A smoother with iteration matrix <IMG
|
||||
WIDTH="32" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img18.png"
|
||||
ALT="$M^k$"> is set up at each level <IMG
|
||||
WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img11.png"
|
||||
ALT="$k < nlev$">, and a solver
|
||||
is set up at the coarsest level, so that they are ready for application
|
||||
(for example, setting up a solver based on the <IMG
|
||||
WIDTH="30" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img19.png"
|
||||
ALT="$LU$"> factorization means computing
|
||||
and storing the <IMG
|
||||
WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img20.png"
|
||||
ALT="$L$"> and <IMG
|
||||
WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$U$"> factors). The construction of the hierarchy of AMG components
|
||||
described so far corresponds to the so-called build phase of the preconditioner.
|
||||
</FONT></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:application_alg"></A><A NAME="524"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
|
||||
Application phase of a V-cycle preconditioner.</CAPTION>
|
||||
<TR><TD>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
$A^k_F = (\bar{a}_{ij}^k)$
|
||||
$\framebox{
|
||||
\begin{minipage}{.85\textwidth}
|
||||
\begin{tabbing}
|
||||
\quad \=\quad \=\quad \=\quad \\[-3mm]
|
||||
procedure V-cycle$\left(k,A^k,b^k,u^k\right)$\ \\[2mm]
|
||||
\>if $\left(k \ne nlev \right)$\ then \\[1mm]
|
||||
\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]
|
||||
\>\> $b^{k+1} = R^{k+1}\left(b^k - A^k u^k\right)$\ \\[1mm]
|
||||
\>\> $u^{k+1} =$\ V-cycle$\left(k+1,A^{k+1},b^{k+1},0\right)$\ \\[1mm]
|
||||
\>\> $u^k = u^k + P^{k+1} u^{k+1}$\ \\[1mm]
|
||||
\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]
|
||||
\>else \\[1mm]
|
||||
\>\> $u^k = \left(A^k\right)^{-1} b^k$\\[1mm]
|
||||
\>endif \\[1mm]
|
||||
\>return $u^k$\ \\[1mm]
|
||||
end
|
||||
\end{tabbing}
|
||||
\end{minipage}
|
||||
}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img42.png"
|
||||
ALT="$A^k_F = (\bar{a}_{ij}^k)$"> is the filtered matrix defined as
|
||||
</FONT></FONT></FONT>
|
||||
<BR>
|
||||
<DIV ALIGN="RIGHT">
|
||||
WIDTH="333" HEIGHT="336" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img22.png"
|
||||
ALT="\framebox{␍\begin{minipage}{.85\textwidth}␍\begin{tabbing}␍\quad \=\quad \=\quad...
|
||||
...mm]␍\>endif [1mm]␍\>return $u^k$ [1mm]␍end␍\end{tabbing}␍\end{minipage}␍}">
|
||||
|
||||
<!-- MATH
|
||||
\begin{equation}
|
||||
\bar{a}_{ij}^k =
|
||||
\left \{ \begin{array}{ll}
|
||||
a_{ij}^k & \mbox{if } j \in \mathcal{N}_i^k(\theta), \\
|
||||
0 & \mbox{otherwise},
|
||||
\end{array} \right.
|
||||
\; (j \ne i),
|
||||
\qquad
|
||||
\bar{a}_{ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),
|
||||
\end{equation}
|
||||
-->
|
||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:filtered"></A><IMG
|
||||
WIDTH="514" HEIGHT="74" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="\begin{displaymath}
|
||||
\bar{a}_{ij}^k =␍ \left \{ \begin{array}{ll}␍ a_{ij}^k & ...
|
||||
...ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),␍\end{displaymath}"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
(5)</TD></TR>
|
||||
</DIV></TD></TR>
|
||||
</TABLE>
|
||||
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
|
||||
and <IMG
|
||||
WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img44.png"
|
||||
ALT="$\omega^k$"> is an approximation of <IMG
|
||||
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img45.png"
|
||||
ALT="$4/(3\rho^k)$">, where
|
||||
<IMG
|
||||
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img46.png"
|
||||
ALT="$\rho^k$"> is the spectral radius of <!-- MATH
|
||||
$(D^k)^{-1}A^k_F$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="83" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img47.png"
|
||||
ALT="$(D^k)^{-1}A^k_F$"> [<A
|
||||
HREF="node29.html#BREZINA_VANEK">2</A>].
|
||||
In MLD2P4 this approximation is obtained by using <!-- MATH
|
||||
$\| A^k_F \|_\infty$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img48.png"
|
||||
ALT="$\Vert A^k_F \Vert _\infty$"> as an estimate
|
||||
of <IMG
|
||||
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img46.png"
|
||||
ALT="$\rho^k$">. Note that for systems coming from uniformly elliptic
|
||||
problems, filtering the matrix <IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img41.png"
|
||||
ALT="$A^k$"> has little or no effect, and
|
||||
<IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img41.png"
|
||||
ALT="$A^k$"> can be used instead of <IMG
|
||||
WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img49.png"
|
||||
ALT="$A^k_F$">. The latter choice is the default in MLD2P4.
|
||||
</DIV>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The components produced in the build phase may be combined in several ways
|
||||
to obtain different multilevel preconditioners;
|
||||
this is done in the application phase, i.e., in the computation of a vector
|
||||
of type <IMG
|
||||
WIDTH="82" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img23.png"
|
||||
ALT="$w=B^{-1}v$">, where <IMG
|
||||
WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img24.png"
|
||||
ALT="$B$"> denotes the preconditioner, usually within an iteration
|
||||
of a Krylov solver [<A
|
||||
HREF="node30.html#Saad_book">20</A>]. An example of such a combination, known as
|
||||
V-cycle, is given in Figure <A HREF="#fig:application_alg">1</A>. In this case, a single iteration
|
||||
of the same smoother is used before and after the the recursive call to the V-cycle (i.e.,
|
||||
in the pre-smoothing and post-smoothing phases); however, different choices can be
|
||||
performed. Other cycles can be defined; in MLD2P4, we implemented the standard V-cycle
|
||||
and W-cycle [<A
|
||||
HREF="node30.html#Briggs2000">3</A>], and a version of the K-cycle described
|
||||
in [<A
|
||||
HREF="node30.html#Notay2008">19</A>].
|
||||
</FONT></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
|
||||
@@ -378,7 +289,7 @@ problems, filtering the matrix <IMG
|
||||
HREF="node14.html">
|
||||
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A>
|
||||
<A NAME="tex2html237"
|
||||
HREF="node11.html">
|
||||
HREF="node12.html">
|
||||
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A>
|
||||
<A NAME="tex2html231"
|
||||
HREF="node12.html">
|
||||
@@ -388,11 +299,11 @@ problems, filtering the matrix <IMG
|
||||
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
|
||||
<BR>
|
||||
<B> Next:</B> <A NAME="tex2html242"
|
||||
HREF="node14.html">Smoothers and coarsest-level solvers</A>
|
||||
HREF="node14.html">Smoothed Aggregation</A>
|
||||
<B> Up:</B> <A NAME="tex2html238"
|
||||
HREF="node11.html">Multigrid Background</A>
|
||||
HREF="node12.html">Multigrid Background</A>
|
||||
<B> Previous:</B> <A NAME="tex2html232"
|
||||
HREF="node12.html">AMG preconditioners</A>
|
||||
HREF="node12.html">Multigrid Background</A>
|
||||
<B> <A NAME="tex2html240"
|
||||
HREF="node2.html">Contents</A></B>
|
||||
<!--End of Navigation Panel-->
|
||||
|
||||
Reference in New Issue
Block a user