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.
317 lines
11 KiB
HTML
317 lines
11 KiB
HTML
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
|
|
|
|
<!--Converted with LaTeX2HTML 2018 (Released Feb 1, 2018) -->
|
|
<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 v2018">
|
|
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
|
|
|
<LINK REL="STYLESHEET" HREF="userhtml.css">
|
|
|
|
<LINK REL="next" HREF="node14.html">
|
|
<LINK REL="previous" HREF="node12.html">
|
|
<LINK REL="up" HREF="node12.html">
|
|
<LINK REL="next" HREF="node14.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
|
|
<DIV CLASS="navigation"><!--Navigation Panel-->
|
|
<A NAME="tex2html249"
|
|
HREF="node14.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A>
|
|
<A NAME="tex2html245"
|
|
HREF="node12.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A>
|
|
<A NAME="tex2html239"
|
|
HREF="node12.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A>
|
|
<A NAME="tex2html247"
|
|
HREF="node2.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html250"
|
|
HREF="node14.html">Smoothed Aggregation</A>
|
|
<B> Up:</B> <A NAME="tex2html246"
|
|
HREF="node12.html">Multigrid Background</A>
|
|
<B> Previous:</B> <A NAME="tex2html240"
|
|
HREF="node12.html">Multigrid Background</A>
|
|
<B> <A NAME="tex2html248"
|
|
HREF="node2.html">Contents</A></B>
|
|
<BR>
|
|
<BR></DIV>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H2><A NAME="SECTION00061000000000000000"></A><A NAME="sec:multilevel"></A>
|
|
<BR>
|
|
AMG preconditioners
|
|
</H2><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
|
<P>
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to describe the AMG preconditioners available in MLD2P4, we consider a
|
|
linear system
|
|
</BIG></BIG></BIG>
|
|
<BR>
|
|
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
|
|
|
|
<!-- MATH
|
|
\begin{equation}
|
|
Ax=b,
|
|
\end{equation}
|
|
-->
|
|
<A NAME="eq:system"></A>
|
|
<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="img2.png"
|
|
ALT="\begin{displaymath}
|
|
Ax=b,
|
|
\end{displaymath}"></TD>
|
|
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
|
(<SPAN CLASS="arabic">2</SPAN>)</TD></TR>
|
|
</TABLE>
|
|
<BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
where <!-- MATH
|
|
$A=(a_{ij}) \in \mathbb{R}^{n \times n}$
|
|
-->
|
|
<SPAN CLASS="MATH"><IMG
|
|
WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img5.png"
|
|
ALT="$A=(a_{ij}) \in \mathbb{R}^{n \times n}$"></SPAN> is a nonsingular sparse matrix;
|
|
for ease of presentation we assume <SPAN CLASS="MATH"><IMG
|
|
WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$A$"></SPAN> has a symmetric sparsity
|
|
pattern.
|
|
</BIG></BIG></BIG>
|
|
<P>
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">Let us consider as finest index space the set of row (column) indices of <SPAN CLASS="MATH"><IMG
|
|
WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$A$"></SPAN>, i.e.,
|
|
<!-- MATH
|
|
$\Omega = \{1, 2, \ldots, n\}$
|
|
-->
|
|
<SPAN CLASS="MATH"><IMG
|
|
WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img6.png"
|
|
ALT="$\Omega = \{1, 2, \ldots, n\}$"></SPAN>.
|
|
Any algebraic multilevel preconditioners implemented in MLD2P4 generates
|
|
a hierarchy of index spaces and a corresponding hierarchy of matrices,
|
|
</BIG></BIG></BIG>
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
|
<!-- 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="img7.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><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
by using the information contained in <SPAN CLASS="MATH"><IMG
|
|
WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$A$"></SPAN>, without assuming any
|
|
knowledge of the geometry of the problem from which <SPAN CLASS="MATH"><IMG
|
|
WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$A$"></SPAN> originates.
|
|
A vector space <!-- MATH
|
|
$\mathbb{R}^{n_{k}}$
|
|
-->
|
|
<SPAN CLASS="MATH"><IMG
|
|
WIDTH="33" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img8.png"
|
|
ALT="$\mathbb{R}^{n_{k}}$"></SPAN> is associated with <SPAN CLASS="MATH"><IMG
|
|
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img9.png"
|
|
ALT="$\Omega^k$"></SPAN>,
|
|
where <SPAN CLASS="MATH"><IMG
|
|
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img10.png"
|
|
ALT="$n_k$"></SPAN> is the size of <SPAN CLASS="MATH"><IMG
|
|
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img9.png"
|
|
ALT="$\Omega^k$"></SPAN>.
|
|
For all <SPAN CLASS="MATH"><IMG
|
|
WIDTH="71" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img11.png"
|
|
ALT="$k < nlev$"></SPAN>, a restriction operator and a prolongation one are built,
|
|
which connect two levels <SPAN CLASS="MATH"><IMG
|
|
WIDTH="14" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img12.png"
|
|
ALT="$k$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
|
WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img13.png"
|
|
ALT="$k+1$"></SPAN>:
|
|
</BIG></BIG></BIG>
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
|
<!-- 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="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><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
the matrix <SPAN CLASS="MATH"><IMG
|
|
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img15.png"
|
|
ALT="$A^{k+1}$"></SPAN> is computed by using the previous operators according
|
|
to the Galerkin approach, i.e.,
|
|
</BIG></BIG></BIG>
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
A^{k+1}=R^kA^kP^k.
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="131" HEIGHT="28" BORDER="0"
|
|
SRC="img16.png"
|
|
ALT="\begin{displaymath}
|
|
A^{k+1}=R^kA^kP^k.
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
In the current implementation of MLD2P4 we have <SPAN CLASS="MATH"><IMG
|
|
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img17.png"
|
|
ALT="$R^k=(P^k)^T$"></SPAN>
|
|
A smoother with iteration matrix <SPAN CLASS="MATH"><IMG
|
|
WIDTH="32" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img18.png"
|
|
ALT="$M^k$"></SPAN> is set up at each level <SPAN CLASS="MATH"><IMG
|
|
WIDTH="71" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img11.png"
|
|
ALT="$k < nlev$"></SPAN>, 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 <SPAN CLASS="MATH"><IMG
|
|
WIDTH="30" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img19.png"
|
|
ALT="$LU$"></SPAN> factorization means computing
|
|
and storing the <SPAN CLASS="MATH"><IMG
|
|
WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img20.png"
|
|
ALT="$L$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
|
WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img21.png"
|
|
ALT="$U$"></SPAN> factors). The construction of the hierarchy of AMG components
|
|
described so far corresponds to the so-called build phase of the preconditioner.
|
|
</BIG></BIG></BIG>
|
|
<P>
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
|
<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
|
|
$\framebox{
|
|
\begin{minipage}{.85\textwidth}
|
|
\begin{tabbing}
|
|
\quad \=\quad \=\quad \=\quad \\[-3mm]
|
|
procedure V-cycle$\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="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}
|
|
}">
|
|
|
|
</DIV></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
|
<P>
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">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 <SPAN CLASS="MATH"><IMG
|
|
WIDTH="82" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img23.png"
|
|
ALT="$w=B^{-1}v$"></SPAN>, where <SPAN CLASS="MATH"><IMG
|
|
WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img24.png"
|
|
ALT="$B$"></SPAN> denotes the preconditioner, usually within an iteration
|
|
of a Krylov solver [<A
|
|
HREF="node36.html#Saad_book">21</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="node36.html#Briggs2000">3</A>], and a version of the K-cycle described
|
|
in [<A
|
|
HREF="node36.html#Notay2008">20</A>].
|
|
</BIG></BIG></BIG>
|
|
<P>
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
|
<DIV CLASS="navigation"><HR>
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html249"
|
|
HREF="node14.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A>
|
|
<A NAME="tex2html245"
|
|
HREF="node12.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A>
|
|
<A NAME="tex2html239"
|
|
HREF="node12.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A>
|
|
<A NAME="tex2html247"
|
|
HREF="node2.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html250"
|
|
HREF="node14.html">Smoothed Aggregation</A>
|
|
<B> Up:</B> <A NAME="tex2html246"
|
|
HREF="node12.html">Multigrid Background</A>
|
|
<B> Previous:</B> <A NAME="tex2html240"
|
|
HREF="node12.html">Multigrid Background</A>
|
|
<B> <A NAME="tex2html248"
|
|
HREF="node2.html">Contents</A></B> </DIV>
|
|
<!--End of Navigation Panel-->
|
|
|
|
</BODY>
|
|
</HTML>
|