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157 lines
5.1 KiB
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157 lines
5.1 KiB
HTML
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original version by: Nikos Drakos, CBLU, University of Leeds
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* with significant contributions from:
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<TITLE>AMG preconditioners</TITLE>
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<H2><A NAME="SECTION00061000000000000000"></A><A NAME="sec:multilevel"></A>
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<BR>
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AMG preconditioners
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</H2>
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In order to describe the AMG preconditioners available in MLD2P4, we consider a
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linear system
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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="img1.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>
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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="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
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SRC="img4.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="img2.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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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="img2.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="img5.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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<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="img6.png"
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ALT="\begin{displaymath}\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}, \end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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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="img2.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="img2.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="img7.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="img8.png"
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ALT="$\Omega^k$">,
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where <IMG
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WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
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SRC="img9.png"
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ALT="$n_k$"> is the size of <IMG
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WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img8.png"
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ALT="$\Omega^k$">.
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For all <IMG
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WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img10.png"
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ALT="$k < nlev$">, a restriction operator and a prolongation one are built,
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which connect two levels <IMG
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WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img11.png"
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ALT="$k$"> and <IMG
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WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img12.png"
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ALT="$k+1$">:
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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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P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad
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R^k \in \mathbb{R}^{n_{k+1}\times n_k};
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\end{displaymath}
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-->
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<IMG
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WIDTH="254" HEIGHT="30" BORDER="0"
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SRC="img13.png"
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ALT="\begin{displaymath}
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P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad
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R^k \in \mathbb{R}^{n_{k+1}\times n_k};
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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