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.
416 lines
14 KiB
HTML
416 lines
14 KiB
HTML
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
|
|
|
|
<!--Converted with LaTeX2HTML 2018 (Released Feb 1, 2018) -->
|
|
<HTML>
|
|
<HEAD>
|
|
<TITLE>Smoothed Aggregation</TITLE>
|
|
<META NAME="description" CONTENT="Smoothed Aggregation">
|
|
<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="node15.html">
|
|
<LINK REL="previous" HREF="node13.html">
|
|
<LINK REL="up" HREF="node12.html">
|
|
<LINK REL="next" HREF="node15.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
|
|
<DIV CLASS="navigation"><!--Navigation Panel-->
|
|
<A NAME="tex2html261"
|
|
HREF="node15.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A>
|
|
<A NAME="tex2html257"
|
|
HREF="node12.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A>
|
|
<A NAME="tex2html251"
|
|
HREF="node13.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A>
|
|
<A NAME="tex2html259"
|
|
HREF="node2.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html262"
|
|
HREF="node15.html">Smoothers and coarsest-level solvers</A>
|
|
<B> Up:</B> <A NAME="tex2html258"
|
|
HREF="node12.html">Multigrid Background</A>
|
|
<B> Previous:</B> <A NAME="tex2html252"
|
|
HREF="node13.html">AMG preconditioners</A>
|
|
<B> <A NAME="tex2html260"
|
|
HREF="node2.html">Contents</A></B>
|
|
<BR>
|
|
<BR></DIV>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H2><A NAME="SECTION00062000000000000000"></A><A NAME="sec:aggregation"></A>
|
|
<BR>
|
|
Smoothed Aggregation
|
|
</H2><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
|
<P>
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to define the prolongator <SPAN CLASS="MATH"><IMG
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img25.png"
|
|
ALT="$P^k$"></SPAN>, used to compute
|
|
the coarse-level matrix <SPAN CLASS="MATH"><IMG
|
|
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img15.png"
|
|
ALT="$A^{k+1}$"></SPAN>, MLD2P4 uses the smoothed aggregation
|
|
algorithm described in [<A
|
|
HREF="node36.html#BREZINA_VANEK">2</A>,<A
|
|
HREF="node36.html#VANEK_MANDEL_BREZINA">26</A>].
|
|
The basic idea of this algorithm is to build a coarse set of indices
|
|
<SPAN CLASS="MATH"><IMG
|
|
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img26.png"
|
|
ALT="$\Omega^{k+1}$"></SPAN> by suitably grouping the indices of <SPAN CLASS="MATH"><IMG
|
|
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img9.png"
|
|
ALT="$\Omega^k$"></SPAN> into disjoint
|
|
subsets (aggregates), and to define the coarse-to-fine space transfer operator
|
|
<SPAN CLASS="MATH"><IMG
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img25.png"
|
|
ALT="$P^k$"></SPAN> by applying a suitable smoother to a simple piecewise constant
|
|
prolongation operator, with the aim of improving the quality of the coarse-space correction.
|
|
</BIG></BIG></BIG>
|
|
<P>
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">Three main steps can be identified in the smoothed aggregation procedure:
|
|
</BIG></BIG></BIG>
|
|
<OL>
|
|
<LI>aggregation of the indices of <SPAN CLASS="MATH"><IMG
|
|
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img9.png"
|
|
ALT="$\Omega^k$"></SPAN> to obtain <SPAN CLASS="MATH"><IMG
|
|
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img26.png"
|
|
ALT="$\Omega^{k+1}$"></SPAN>;
|
|
</LI>
|
|
<LI>construction of the prolongator <SPAN CLASS="MATH"><IMG
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img25.png"
|
|
ALT="$P^k$"></SPAN>;
|
|
</LI>
|
|
<LI>application of <SPAN CLASS="MATH"><IMG
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img25.png"
|
|
ALT="$P^k$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
|
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img17.png"
|
|
ALT="$R^k=(P^k)^T$"></SPAN> to build <SPAN CLASS="MATH"><IMG
|
|
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img15.png"
|
|
ALT="$A^{k+1}$"></SPAN>.
|
|
</LI>
|
|
</OL><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
|
<P>
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to perform the coarsening step, the smoothed aggregation algorithm
|
|
described in [<A
|
|
HREF="node36.html#VANEK_MANDEL_BREZINA">26</A>] is used. In this algorithm,
|
|
each index <!-- MATH
|
|
$j \in \Omega^{k+1}$
|
|
-->
|
|
<SPAN CLASS="MATH"><IMG
|
|
WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img27.png"
|
|
ALT="$j \in \Omega^{k+1}$"></SPAN> corresponds to an aggregate <SPAN CLASS="MATH"><IMG
|
|
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img28.png"
|
|
ALT="$\Omega^k_j$"></SPAN> of <SPAN CLASS="MATH"><IMG
|
|
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img9.png"
|
|
ALT="$\Omega^k$"></SPAN>,
|
|
consisting of a suitably chosen index <!-- MATH
|
|
$i \in \Omega^k$
|
|
-->
|
|
<SPAN CLASS="MATH"><IMG
|
|
WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img29.png"
|
|
ALT="$i \in \Omega^k$"></SPAN> and indices that are (usually) contained in a
|
|
strongly-coupled neighborood of <SPAN CLASS="MATH"><IMG
|
|
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img30.png"
|
|
ALT="$i$"></SPAN>, i.e.,
|
|
</BIG></BIG></BIG>
|
|
<BR>
|
|
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
|
|
|
|
<!-- MATH
|
|
\begin{equation}
|
|
\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
|
|
\left\{ r \in \Omega^k: |a_{ir}^k| > \theta \sqrt{|a_{ii}^ka_{rr}^k|} \right \} \cup \left\{ i \right\},
|
|
\end{equation}
|
|
-->
|
|
<A NAME="eq:strongly_coup"></A>
|
|
<TABLE WIDTH="100%" ALIGN="CENTER">
|
|
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:strongly_coup"></A><IMG
|
|
WIDTH="387" HEIGHT="48" BORDER="0"
|
|
SRC="img31.png"
|
|
ALT="\begin{displaymath}
|
|
\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
|
|
\left\{ r \i...
|
|
...vert a_{ii}^ka_{rr}^k\vert} \right \} \cup \left\{ i \right\},
|
|
\end{displaymath}"></TD>
|
|
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
|
(<SPAN CLASS="arabic">3</SPAN>)</TD></TR>
|
|
</TABLE>
|
|
<BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
for a given threshold <!-- MATH
|
|
$\theta \in [0,1]$
|
|
-->
|
|
<SPAN CLASS="MATH"><IMG
|
|
WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img32.png"
|
|
ALT="$\theta \in [0,1]$"></SPAN> (see [<A
|
|
HREF="node36.html#VANEK_MANDEL_BREZINA">26</A>] for the details).
|
|
Since this algorithm has a sequential nature, a decoupled
|
|
version of it is applied, where each processor independently executes
|
|
the algorithm on the set of indices assigned to it in the initial data
|
|
distribution. This version is embarrassingly parallel, since it does not require any data
|
|
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
|
|
of the matrix <SPAN CLASS="MATH"><IMG
|
|
WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$A$"></SPAN>. Nevertheless, this parallel algorithm has been chosen for
|
|
MLD2P4, since it has been shown to produce good results in practice
|
|
[<A
|
|
HREF="node36.html#aaecc_07">5</A>,<A
|
|
HREF="node36.html#apnum_07">7</A>,<A
|
|
HREF="node36.html#TUMINARO_TONG">25</A>].
|
|
</BIG></BIG></BIG>
|
|
<P>
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The prolongator <SPAN CLASS="MATH"><IMG
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img25.png"
|
|
ALT="$P^k$"></SPAN> is built starting from a tentative prolongator
|
|
<!-- MATH
|
|
$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$
|
|
-->
|
|
<SPAN CLASS="MATH"><IMG
|
|
WIDTH="117" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img33.png"
|
|
ALT="$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$"></SPAN>, defined as
|
|
</BIG></BIG></BIG>
|
|
<BR>
|
|
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
|
|
|
|
<!-- MATH
|
|
\begin{equation}
|
|
\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
|
|
\left\{ \begin{array}{ll}
|
|
1 & \quad \mbox{if} \; i \in \Omega^k_j, \\
|
|
0 & \quad \mbox{otherwise},
|
|
\end{array} \right.
|
|
\end{equation}
|
|
-->
|
|
<A NAME="eq:tent_prol"></A>
|
|
<TABLE WIDTH="100%" ALIGN="CENTER">
|
|
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:tent_prol"></A><IMG
|
|
WIDTH="286" HEIGHT="51" BORDER="0"
|
|
SRC="img34.png"
|
|
ALT="\begin{displaymath}
|
|
\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
|
|
\left\{...
|
|
...Omega^k_j, \\
|
|
0 & \quad \mbox{otherwise},
|
|
\end{array} \right.
|
|
\end{displaymath}"></TD>
|
|
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
|
(<SPAN CLASS="arabic">4</SPAN>)</TD></TR>
|
|
</TABLE>
|
|
<BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
where <SPAN CLASS="MATH"><IMG
|
|
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img28.png"
|
|
ALT="$\Omega^k_j$"></SPAN> is the aggregate of <SPAN CLASS="MATH"><IMG
|
|
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img9.png"
|
|
ALT="$\Omega^k$"></SPAN>
|
|
corresponding to the index <!-- MATH
|
|
$j \in \Omega^{k+1}$
|
|
-->
|
|
<SPAN CLASS="MATH"><IMG
|
|
WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img27.png"
|
|
ALT="$j \in \Omega^{k+1}$"></SPAN>.
|
|
<SPAN CLASS="MATH"><IMG
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img25.png"
|
|
ALT="$P^k$"></SPAN> is obtained by applying to <SPAN CLASS="MATH"><IMG
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img35.png"
|
|
ALT="$\bar{P}^k$"></SPAN> a smoother
|
|
<!-- MATH
|
|
$S^k \in \mathbb{R}^{n_k \times n_k}$
|
|
-->
|
|
<SPAN CLASS="MATH"><IMG
|
|
WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img36.png"
|
|
ALT="$S^k \in \mathbb{R}^{n_k \times n_k}$"></SPAN>:
|
|
</BIG></BIG></BIG>
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
P^k = S^k \bar{P}^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}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
in order to remove nonsmooth components from the range of the prolongator,
|
|
and hence to improve the convergence properties of the multilevel
|
|
method [<A
|
|
HREF="node36.html#BREZINA_VANEK">2</A>,<A
|
|
HREF="node36.html#Stuben_01">24</A>].
|
|
A simple choice for <SPAN CLASS="MATH"><IMG
|
|
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img38.png"
|
|
ALT="$S^k$"></SPAN> is the damped Jacobi smoother:
|
|
</BIG></BIG></BIG>
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
S^k = I - \omega^k (D^k)^{-1} A^k_F ,
|
|
\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}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
where <SPAN CLASS="MATH"><IMG
|
|
WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img40.png"
|
|
ALT="$D^k$"></SPAN> is the diagonal matrix with the same diagonal entries as <SPAN CLASS="MATH"><IMG
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img41.png"
|
|
ALT="$A^k$"></SPAN>,
|
|
<!-- MATH
|
|
$A^k_F = (\bar{a}_{ij}^k)$
|
|
-->
|
|
<SPAN CLASS="MATH"><IMG
|
|
WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img42.png"
|
|
ALT="$A^k_F = (\bar{a}_{ij}^k)$"></SPAN> is the filtered matrix defined as
|
|
</BIG></BIG></BIG>
|
|
<BR>
|
|
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
|
|
|
|
<!-- 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}
|
|
-->
|
|
<A NAME="eq:filtered"></A>
|
|
<TABLE WIDTH="100%" ALIGN="CENTER">
|
|
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:filtered"></A><IMG
|
|
WIDTH="499" HEIGHT="59" BORDER="0"
|
|
SRC="img43.png"
|
|
ALT="\begin{displaymath}
|
|
\bar{a}_{ij}^k =
|
|
\left \{ \begin{array}{ll}
|
|
a_{ij}^k & \m...
|
|
...ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),
|
|
\end{displaymath}"></TD>
|
|
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
|
(<SPAN CLASS="arabic">5</SPAN>)</TD></TR>
|
|
</TABLE>
|
|
<BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
and <SPAN CLASS="MATH"><IMG
|
|
WIDTH="24" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img44.png"
|
|
ALT="$\omega^k$"></SPAN> is an approximation of <SPAN CLASS="MATH"><IMG
|
|
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img45.png"
|
|
ALT="$4/(3\rho^k)$"></SPAN>, where
|
|
<SPAN CLASS="MATH"><IMG
|
|
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img46.png"
|
|
ALT="$\rho^k$"></SPAN> is the spectral radius of <!-- MATH
|
|
$(D^k)^{-1}A^k_F$
|
|
-->
|
|
<SPAN CLASS="MATH"><IMG
|
|
WIDTH="83" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img47.png"
|
|
ALT="$(D^k)^{-1}A^k_F$"></SPAN> [<A
|
|
HREF="node36.html#BREZINA_VANEK">2</A>].
|
|
In MLD2P4 this approximation is obtained by using <!-- MATH
|
|
$\| A^k_F \|_\infty$
|
|
-->
|
|
<SPAN CLASS="MATH"><IMG
|
|
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img48.png"
|
|
ALT="$\Vert A^k_F \Vert _\infty$"></SPAN> as an estimate
|
|
of <SPAN CLASS="MATH"><IMG
|
|
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img46.png"
|
|
ALT="$\rho^k$"></SPAN>. Note that for systems coming from uniformly elliptic
|
|
problems, filtering the matrix <SPAN CLASS="MATH"><IMG
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img41.png"
|
|
ALT="$A^k$"></SPAN> has little or no effect, and
|
|
<SPAN CLASS="MATH"><IMG
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img41.png"
|
|
ALT="$A^k$"></SPAN> can be used instead of <SPAN CLASS="MATH"><IMG
|
|
WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img49.png"
|
|
ALT="$A^k_F$"></SPAN>. The latter choice is the default in MLD2P4.
|
|
</BIG></BIG></BIG>
|
|
<P>
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
|
<DIV CLASS="navigation"><HR>
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html261"
|
|
HREF="node15.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A>
|
|
<A NAME="tex2html257"
|
|
HREF="node12.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A>
|
|
<A NAME="tex2html251"
|
|
HREF="node13.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A>
|
|
<A NAME="tex2html259"
|
|
HREF="node2.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html262"
|
|
HREF="node15.html">Smoothers and coarsest-level solvers</A>
|
|
<B> Up:</B> <A NAME="tex2html258"
|
|
HREF="node12.html">Multigrid Background</A>
|
|
<B> Previous:</B> <A NAME="tex2html252"
|
|
HREF="node13.html">AMG preconditioners</A>
|
|
<B> <A NAME="tex2html260"
|
|
HREF="node2.html">Contents</A></B> </DIV>
|
|
<!--End of Navigation Panel-->
|
|
|
|
</BODY>
|
|
</HTML>
|