You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
157 lines
5.1 KiB
HTML
157 lines
5.1 KiB
HTML
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
|
|
|
|
<!--Converted with LaTeX2HTML 2012 (1.2)
|
|
original version by: Nikos Drakos, CBLU, University of Leeds
|
|
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
|
|
* with significant contributions from:
|
|
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
|
|
<HTML>
|
|
<HEAD>
|
|
<TITLE>AMG preconditioners</TITLE>
|
|
<META NAME="description" CONTENT="AMG preconditioners">
|
|
<META NAME="keywords" CONTENT="userhtml">
|
|
<META NAME="resource-type" CONTENT="document">
|
|
<META NAME="distribution" CONTENT="global">
|
|
|
|
<META NAME="Generator" CONTENT="LaTeX2HTML v2012">
|
|
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
|
|
|
<LINK REL="STYLESHEET" HREF="userhtml.css">
|
|
|
|
<LINK REL="previous" HREF="node11.html">
|
|
<LINK REL="up" HREF="node11.html">
|
|
<LINK REL="next" HREF="node13.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html218"
|
|
HREF="node13.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A>
|
|
<A NAME="tex2html214"
|
|
HREF="node11.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A>
|
|
<A NAME="tex2html210"
|
|
HREF="node11.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A>
|
|
<A NAME="tex2html216"
|
|
HREF="node2.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html219"
|
|
HREF="node13.html">Getting Started</A>
|
|
<B> Up:</B> <A NAME="tex2html215"
|
|
HREF="node11.html">Multigrid Background</A>
|
|
<B> Previous:</B> <A NAME="tex2html211"
|
|
HREF="node11.html">Multigrid Background</A>
|
|
<B> <A NAME="tex2html217"
|
|
HREF="node2.html">Contents</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H2><A NAME="SECTION00061000000000000000"></A><A NAME="sec:multilevel"></A>
|
|
<BR>
|
|
AMG preconditioners
|
|
</H2>
|
|
|
|
In order to describe the AMG preconditioners available in MLD2P4, we consider a
|
|
linear system
|
|
<BR>
|
|
<DIV ALIGN="RIGHT">
|
|
|
|
<!-- MATH
|
|
\begin{equation}
|
|
Ax=b,
|
|
\end{equation}
|
|
-->
|
|
<TABLE WIDTH="100%" ALIGN="CENTER">
|
|
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:system"></A><IMG
|
|
WIDTH="58" HEIGHT="30" BORDER="0"
|
|
SRC="img1.png"
|
|
ALT="\begin{displaymath}
|
|
Ax=b,
|
|
\end{displaymath}"></TD>
|
|
<TD WIDTH=10 ALIGN="RIGHT">
|
|
(2)</TD></TR>
|
|
</TABLE>
|
|
<BR CLEAR="ALL"></DIV><P></P>
|
|
where <!-- MATH
|
|
$A=(a_{ij}) \in \mathbb{R}^{n \times n}$
|
|
-->
|
|
<IMG
|
|
WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img4.png"
|
|
ALT="$A=(a_{ij}) \in \mathbb{R}^{n \times n}$"> is a nonsingular sparse matrix;
|
|
for ease of presentation we assume <IMG
|
|
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img2.png"
|
|
ALT="$A$"> is real, but the
|
|
results are valid for the complex case as well.
|
|
|
|
Let us assume as finest index space the set of row (column) indices of <IMG
|
|
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img2.png"
|
|
ALT="$A$">, i.e.,
|
|
<!-- MATH
|
|
$\Omega = \{1, 2, \ldots, n\}$
|
|
-->
|
|
<IMG
|
|
WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img5.png"
|
|
ALT="$\Omega = \{1, 2, \ldots, n\}$">.
|
|
Any algebraic multilevel preconditioners implemented in MLD2P4 generates
|
|
a hierarchy of index spaces and a corresponding hierarchy of matrices,
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\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}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="398" HEIGHT="30" BORDER="0"
|
|
SRC="img6.png"
|
|
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}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
by using the information contained in <IMG
|
|
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img2.png"
|
|
ALT="$A$">, without assuming any
|
|
knowledge of the geometry of the problem from which <IMG
|
|
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img2.png"
|
|
ALT="$A$"> originates.
|
|
A vector space <!-- MATH
|
|
$\mathbb{R}^{n_{k}}$
|
|
-->
|
|
<IMG
|
|
WIDTH="33" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img7.png"
|
|
ALT="$\mathbb{R}^{n_{k}}$"> is associated with <IMG
|
|
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img8.png"
|
|
ALT="$\Omega^k$">,
|
|
where <IMG
|
|
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img9.png"
|
|
ALT="$n_k$"> is the size of <IMG
|
|
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img8.png"
|
|
ALT="$\Omega^k$">.
|
|
For all <IMG
|
|
WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img10.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="img11.png"
|
|
ALT="$k$"> and <IMG
|
|
WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img12.png"
|
|
ALT="$k+1$">:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\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}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="254" HEIGHT="30" BORDER="0"
|
|
SRC="img13.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>
|
|
<BR><HR>
|
|
|
|
</BODY>
|
|
</HTML>
|