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.
318 lines
9.6 KiB
HTML
318 lines
9.6 KiB
HTML
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
|
|
|
|
<!--Converted with LaTeX2HTML 2008 (1.71)
|
|
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>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 v2008">
|
|
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
|
|
|
<LINK REL="STYLESHEET" HREF="userhtml.css">
|
|
|
|
<LINK REL="previous" HREF="node12.html">
|
|
<LINK REL="up" HREF="node11.html">
|
|
<LINK REL="next" HREF="node14.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html219"
|
|
HREF="node14.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A>
|
|
<A NAME="tex2html215"
|
|
HREF="node11.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A>
|
|
<A NAME="tex2html211"
|
|
HREF="node12.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A>
|
|
<A NAME="tex2html217"
|
|
HREF="node2.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html220"
|
|
HREF="node14.html">Getting Started</A>
|
|
<B> Up:</B> <A NAME="tex2html216"
|
|
HREF="node11.html">Multi-level Domain Decomposition Background</A>
|
|
<B> Previous:</B> <A NAME="tex2html212"
|
|
HREF="node12.html">Multi-level Schwarz Preconditioners</A>
|
|
<B> <A NAME="tex2html218"
|
|
HREF="node2.html">Contents</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H2><A NAME="SECTION00062000000000000000"></A><A NAME="sec:aggregation"></A>
|
|
<BR>
|
|
Smoothed Aggregation
|
|
</H2>
|
|
|
|
<P>
|
|
In order to define the restriction operator <IMG
|
|
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img69.png"
|
|
ALT="$R_C$">, which is used to compute
|
|
the coarse-level matrix <IMG
|
|
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img44.png"
|
|
ALT="$A_C$">, MLD2P4 uses the <I>smoothed aggregation</I>
|
|
algorithm described in [<A
|
|
HREF="node25.html#BREZINA_VANEK">1</A>,<A
|
|
HREF="node25.html#VANEK_MANDEL_BREZINA">26</A>].
|
|
The basic idea of this algorithm is to build a coarse set of vertices
|
|
<IMG
|
|
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img45.png"
|
|
ALT="$W_C$"> by suitably grouping the vertices of <IMG
|
|
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img11.png"
|
|
ALT="$W$"> into disjoint subsets
|
|
(aggregates), and to define the coarse-to-fine space transfer operator <IMG
|
|
WIDTH="29" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img70.png"
|
|
ALT="$R_C^T$"> by
|
|
applying a suitable smoother to a simple piecewise constant
|
|
prolongation operator, to improve the quality of the coarse-space correction.
|
|
|
|
<P>
|
|
Three main steps can be identified in the smoothed aggregation procedure:
|
|
|
|
<OL>
|
|
<LI>coarsening of the vertex set <IMG
|
|
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img11.png"
|
|
ALT="$W$">, to obtain <IMG
|
|
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img45.png"
|
|
ALT="$W_C$">;
|
|
</LI>
|
|
<LI>construction of the prolongator <IMG
|
|
WIDTH="29" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img70.png"
|
|
ALT="$R_C^T$">;
|
|
</LI>
|
|
<LI>application of <IMG
|
|
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img69.png"
|
|
ALT="$R_C$"> and <IMG
|
|
WIDTH="29" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img70.png"
|
|
ALT="$R_C^T$"> to build <IMG
|
|
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img44.png"
|
|
ALT="$A_C$">.
|
|
</LI>
|
|
</OL>
|
|
|
|
<P>
|
|
To perform the coarsening step, we have implemented the aggregation algorithm sketched
|
|
in [<A
|
|
HREF="node25.html#apnum_07">4</A>]. According to [<A
|
|
HREF="node25.html#VANEK_MANDEL_BREZINA">26</A>], a modification of
|
|
this algorithm has been actually considered,
|
|
in which each aggregate <IMG
|
|
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img71.png"
|
|
ALT="$N_r$"> is made of vertices of <IMG
|
|
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img11.png"
|
|
ALT="$W$"> that are <I>strongly coupled</I>
|
|
to a certain root vertex <IMG
|
|
WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img72.png"
|
|
ALT="$r \in W$">, i.e. <BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
N_r = \left\{s \in W: |a_{rs}| > \theta \sqrt{|a_{rr}a_{ss}|} \right\}
|
|
\cup \left\{ r \right\} ,
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="319" HEIGHT="38" BORDER="0"
|
|
SRC="img73.png"
|
|
ALT="\begin{displaymath}N_r = \left\{s \in W: \vert a_{rs}\vert > \theta \sqrt{\vert a_{rr}a_{ss}\vert} \right\}
|
|
\cup \left\{ r \right\} ,
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
for a given <!-- MATH
|
|
$\theta \in [0,1]$
|
|
-->
|
|
<IMG
|
|
WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img74.png"
|
|
ALT="$\theta \in [0,1]$">.
|
|
Since this algorithm has a sequential nature, a <I>decoupled</I> version of
|
|
it has been chosen, where each processor <IMG
|
|
WIDTH="10" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img75.png"
|
|
ALT="$i$"> independently applies the algorithm to
|
|
the set of vertices <IMG
|
|
WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img76.png"
|
|
ALT="$W_i^0$"> 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 non-uniform aggregates near boundary vertices,
|
|
i.e. near vertices adjacent to vertices in other processors, and is strongly
|
|
dependent on the number of processors and on the initial partitioning of the matrix <IMG
|
|
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img2.png"
|
|
ALT="$A$">.
|
|
Nevertheless, this algorithm has been chosen for the implementation in MLD2P4,
|
|
since it has been shown to produce good results in practice
|
|
[<A
|
|
HREF="node25.html#aaecc_07">3</A>,<A
|
|
HREF="node25.html#apnum_07">4</A>,<A
|
|
HREF="node25.html#TUMINARO_TONG">25</A>].
|
|
|
|
<P>
|
|
The prolongator <IMG
|
|
WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img77.png"
|
|
ALT="$P_C=R_C^T$"> is built starting from a <I>tentative prolongator</I>
|
|
<!-- MATH
|
|
$P \in \Re^{n \times n_C}$
|
|
-->
|
|
<IMG
|
|
WIDTH="90" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img78.png"
|
|
ALT="$P \in \Re^{n \times n_C}$">, defined as
|
|
<BR>
|
|
<DIV ALIGN="RIGHT">
|
|
|
|
<!-- MATH
|
|
\begin{equation}
|
|
P=(p_{ij}), \quad p_{ij}=
|
|
\left\{ \begin{array}{ll}
|
|
1 & \quad \mbox{if} \; i \in V^j_C \\
|
|
0 & \quad \mbox{otherwise}
|
|
\end{array} \right. .
|
|
\end{equation}
|
|
-->
|
|
<TABLE WIDTH="100%" ALIGN="CENTER">
|
|
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:tent_prol"></A><IMG
|
|
WIDTH="290" HEIGHT="52" BORDER="0"
|
|
SRC="img79.png"
|
|
ALT="\begin{displaymath}
|
|
P=(p_{ij}), \quad p_{ij}=
|
|
\left\{ \begin{array}{ll}
|
|
1 & \qu...
|
|
...\in V^j_C \\
|
|
0 & \quad \mbox{otherwise}
|
|
\end{array} \right. .
|
|
\end{displaymath}"></TD>
|
|
<TD WIDTH=10 ALIGN="RIGHT">
|
|
(2)</TD></TR>
|
|
</TABLE>
|
|
<BR CLEAR="ALL"></DIV><P></P>
|
|
<IMG
|
|
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img80.png"
|
|
ALT="$P_C$"> is obtained by
|
|
applying to <IMG
|
|
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img81.png"
|
|
ALT="$P$"> a smoother <!-- MATH
|
|
$S \in \Re^{n \times n}$
|
|
-->
|
|
<IMG
|
|
WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img82.png"
|
|
ALT="$S \in \Re^{n \times n}$">:
|
|
<BR>
|
|
<DIV ALIGN="RIGHT">
|
|
|
|
<!-- MATH
|
|
\begin{equation}
|
|
P_C = S P,
|
|
\end{equation}
|
|
-->
|
|
<TABLE WIDTH="100%" ALIGN="CENTER">
|
|
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:smoothed_prol"></A><IMG
|
|
WIDTH="73" HEIGHT="30" BORDER="0"
|
|
SRC="img83.png"
|
|
ALT="\begin{displaymath}
|
|
P_C = S P,
|
|
\end{displaymath}"></TD>
|
|
<TD WIDTH=10 ALIGN="RIGHT">
|
|
(3)</TD></TR>
|
|
</TABLE>
|
|
<BR CLEAR="ALL"></DIV><P></P>
|
|
in order to remove oscillatory components from the range of the prolongator
|
|
and hence to improve the convergence properties of the multi-level
|
|
Schwarz method [<A
|
|
HREF="node25.html#BREZINA_VANEK">1</A>,<A
|
|
HREF="node25.html#StubenGMD69_99">24</A>].
|
|
A simple choice for <IMG
|
|
WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img84.png"
|
|
ALT="$S$"> is the damped Jacobi smoother:
|
|
<BR>
|
|
<DIV ALIGN="RIGHT">
|
|
|
|
<!-- MATH
|
|
\begin{equation}
|
|
S = I - \omega D^{-1} A ,
|
|
\end{equation}
|
|
-->
|
|
<TABLE WIDTH="100%" ALIGN="CENTER">
|
|
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:jac_smoother"></A><IMG
|
|
WIDTH="126" HEIGHT="30" BORDER="0"
|
|
SRC="img85.png"
|
|
ALT="\begin{displaymath}
|
|
S = I - \omega D^{-1} A ,
|
|
\end{displaymath}"></TD>
|
|
<TD WIDTH=10 ALIGN="RIGHT">
|
|
(4)</TD></TR>
|
|
</TABLE>
|
|
<BR CLEAR="ALL"></DIV><P></P>
|
|
where the value of <IMG
|
|
WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img86.png"
|
|
ALT="$\omega$"> can be chosen
|
|
using some estimate of the spectral radius of <IMG
|
|
WIDTH="50" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img87.png"
|
|
ALT="$D^{-1}A$"> [<A
|
|
HREF="node25.html#BREZINA_VANEK">1</A>].
|
|
|
|
<P>
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html219"
|
|
HREF="node14.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A>
|
|
<A NAME="tex2html215"
|
|
HREF="node11.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A>
|
|
<A NAME="tex2html211"
|
|
HREF="node12.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A>
|
|
<A NAME="tex2html217"
|
|
HREF="node2.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html220"
|
|
HREF="node14.html">Getting Started</A>
|
|
<B> Up:</B> <A NAME="tex2html216"
|
|
HREF="node11.html">Multi-level Domain Decomposition Background</A>
|
|
<B> Previous:</B> <A NAME="tex2html212"
|
|
HREF="node12.html">Multi-level Schwarz Preconditioners</A>
|
|
<B> <A NAME="tex2html218"
|
|
HREF="node2.html">Contents</A></B>
|
|
<!--End of Navigation Panel-->
|
|
|
|
</BODY>
|
|
</HTML>
|