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mld2p4:

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Further doc fixes.
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Salvatore Filippone
2008-07-24 06:05:46 +00:00
parent 4dda0494a7
commit 78805bf653
130 changed files with 10569 additions and 7036 deletions
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@@ -1,4 +1,4 @@
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
original version by: Nikos Drakos, CBLU, University of Leeds
@@ -24,8 +24,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds
</HEAD>
<BODY >
<DIV CLASS="navigation"><!--Navigation Panel-->
<!--Navigation Panel-->
<A NAME="tex2html204"
HREF="node13.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
@@ -52,7 +51,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds
&nbsp; <B> <A NAME="tex2html203"
HREF="node2.html">Contents</A></B>
<BR>
<BR></DIV>
<BR>
<!--End of Navigation Panel-->
<H2><A NAME="SECTION00062000000000000000"></A><A NAME="sec:aggregation"></A>
@@ -61,29 +60,29 @@ Smoothed Aggregation
</H2>
<P>
In order to define the restriction operator <SPAN CLASS="MATH"><IMG
In order to define the restriction operator <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img67.png"
ALT="$R_C$"></SPAN>, which is used to compute
the coarse-level matrix <SPAN CLASS="MATH"><IMG
ALT="$R_C$">, which is used to compute
the coarse-level matrix <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img43.png"
ALT="$A_C$"></SPAN>, MLD2P4 uses the <SPAN CLASS="textit">smoothed aggregation</SPAN>
ALT="$A_C$">, MLD2P4 uses the <I>smoothed aggregation</I>
algorithm described in [<A
HREF="node24.html#BREZINA_VANEK">1</A>,<A
HREF="node24.html#VANEK_MANDEL_BREZINA">24</A>].
The basic idea of this algorithm is to build a coarse set of vertices
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img44.png"
ALT="$W_C$"></SPAN> by suitably grouping the vertices of <SPAN CLASS="MATH"><IMG
ALT="$W_C$"> by suitably grouping the vertices of <IMG
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img10.png"
ALT="$W$"></SPAN> into disjoint subsets
(aggregates), and to define the coarse-to-fine space transfer operator <SPAN CLASS="MATH"><IMG
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="img68.png"
ALT="$R_C^T$"></SPAN> by
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.
@@ -91,29 +90,29 @@ prolongation operator, to improve the quality of the coarse-space correction.
Three main steps can be identified in the smoothed aggregation procedure:
<OL>
<LI>coarsening of the vertex set <SPAN CLASS="MATH"><IMG
<LI>coarsening of the vertex set <IMG
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img10.png"
ALT="$W$"></SPAN>, to obtain <SPAN CLASS="MATH"><IMG
ALT="$W$">, to obtain <IMG
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img44.png"
ALT="$W_C$"></SPAN>;
ALT="$W_C$">;
</LI>
<LI>construction of the prolongator <SPAN CLASS="MATH"><IMG
<LI>construction of the prolongator <IMG
WIDTH="29" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
SRC="img68.png"
ALT="$R_C^T$"></SPAN>;
ALT="$R_C^T$">;
</LI>
<LI>application of <SPAN CLASS="MATH"><IMG
<LI>application of <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img67.png"
ALT="$R_C$"></SPAN> and <SPAN CLASS="MATH"><IMG
ALT="$R_C$"> and <IMG
WIDTH="29" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
SRC="img68.png"
ALT="$R_C^T$"></SPAN> to build <SPAN CLASS="MATH"><IMG
ALT="$R_C^T$"> to build <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img43.png"
ALT="$A_C$"></SPAN>.
ALT="$A_C$">.
</LI>
</OL>
@@ -123,18 +122,18 @@ in [<A
HREF="node24.html#apnum_07">4</A>]. According to [<A
HREF="node24.html#VANEK_MANDEL_BREZINA">24</A>], a modification of
this algorithm has been actually considered,
in which each aggregate <SPAN CLASS="MATH"><IMG
in which each aggregate <IMG
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img69.png"
ALT="$N_r$"></SPAN> is made of vertices of <SPAN CLASS="MATH"><IMG
ALT="$N_r$"> is made of vertices of <IMG
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img10.png"
ALT="$W$"></SPAN> that are <SPAN CLASS="textit">strongly coupled</SPAN>
to a certain root vertex <SPAN CLASS="MATH"><IMG
ALT="$W$"> that are <I>strongly coupled</I>
to a certain root vertex <IMG
WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img70.png"
ALT="$r \in W$"></SPAN>, i.e. <BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
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\}
@@ -143,7 +142,7 @@ N_r = \left\{s \in W: |a_{rs}| > \theta \sqrt{|a_{rr}a_{ss}|} \right\}
-->
<IMG
WIDTH="319" HEIGHT="38" BORDER="0"
WIDTH="320" HEIGHT="38" BORDER="0"
SRC="img71.png"
ALT="\begin{displaymath}N_r = \left\{s \in W: \vert a_{rs}\vert &gt; \theta \sqrt{\vert a_{rr}a_{ss}\vert} \right\}
\cup \left\{ r \right\} ,
@@ -154,26 +153,26 @@ N_r = \left\{s \in W: |a_{rs}| > \theta \sqrt{|a_{rr}a_{ss}|} \right\}
for a given <!-- MATH
$\theta \in [0,1]$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img72.png"
ALT="$\theta \in [0,1]$"></SPAN>.
Since this algorithm has a sequential nature, a <SPAN CLASS="textit">decoupled</SPAN> version of
it has been chosen, where each processor <SPAN CLASS="MATH"><IMG
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="img73.png"
ALT="$i$"></SPAN> independently applies the algorithm to
the set of vertices <SPAN CLASS="MATH"><IMG
ALT="$i$"> independently applies the algorithm to
the set of vertices <IMG
WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img74.png"
ALT="$W_i^0$"></SPAN> assigned to it in the initial data distribution. This
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 <SPAN CLASS="MATH"><IMG
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="$A$"></SPAN>.
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
@@ -182,19 +181,19 @@ since it has been shown to produce good results in practice
HREF="node24.html#TUMINARO_TONG">23</A>].
<P>
The prolongator <SPAN CLASS="MATH"><IMG
The prolongator <IMG
WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
SRC="img75.png"
ALT="$P_C=R_C^T$"></SPAN> is built starting from a <SPAN CLASS="textit">tentative prolongator</SPAN>
ALT="$P_C=R_C^T$"> is built starting from a <I>tentative prolongator</I>
<!-- MATH
$P \in \Re^{n \times n_C}$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="90" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img76.png"
ALT="$P \in \Re^{n \times n_C}$"></SPAN>, defined as
ALT="$P \in \Re^{n \times n_C}$">, defined as
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
<DIV ALIGN="RIGHT">
<!-- MATH
\begin{equation}
@@ -205,7 +204,6 @@ P=(p_{ij}), \quad p_{ij}=
\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="290" HEIGHT="52" BORDER="0"
@@ -218,33 +216,32 @@ P=(p_{ij}), \quad p_{ij}=
0 &amp; \quad \mbox{otherwise}
\end{array} \right. .
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">2</SPAN>)</TD></TR>
<TD WIDTH=10 ALIGN="RIGHT">
(2)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img78.png"
ALT="$P_C$"></SPAN> is obtained by
applying to <SPAN CLASS="MATH"><IMG
ALT="$P_C$"> is obtained by
applying to <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img79.png"
ALT="$P$"></SPAN> a smoother <!-- MATH
ALT="$P$"> a smoother <!-- MATH
$S \in \Re^{n \times n}$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img80.png"
ALT="$S \in \Re^{n \times n}$"></SPAN>:
ALT="$S \in \Re^{n \times n}$">:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
<DIV ALIGN="RIGHT">
<!-- MATH
\begin{equation}
P_C = S P,
\end{equation}
-->
<A NAME="eq:smoothed_prol"></A>
<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"
@@ -252,8 +249,8 @@ P_C = S P,
ALT="\begin{displaymath}
P_C = S P,
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">3</SPAN>)</TD></TR>
<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
@@ -261,43 +258,41 @@ and hence to improve the convergence properties of the multi-level
Schwarz method [<A
HREF="node24.html#BREZINA_VANEK">1</A>,<A
HREF="node24.html#StubenGMD69_99">22</A>].
A simple choice for <SPAN CLASS="MATH"><IMG
A simple choice for <IMG
WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img82.png"
ALT="$S$"></SPAN> is the damped Jacobi smoother:
ALT="$S$"> is the damped Jacobi smoother:
<BR>
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
<DIV ALIGN="RIGHT">
<!-- MATH
\begin{equation}
S = I - \omega D^{-1} A ,
\end{equation}
-->
<A NAME="eq:jac_smoother"></A>
<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"
WIDTH="125" HEIGHT="30" BORDER="0"
SRC="img83.png"
ALT="\begin{displaymath}
S = I - \omega D^{-1} A ,
\end{displaymath}"></TD>
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
(<SPAN CLASS="arabic">4</SPAN>)</TD></TR>
<TD WIDTH=10 ALIGN="RIGHT">
(4)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
where the value of <SPAN CLASS="MATH"><IMG
where the value of <IMG
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img84.png"
ALT="$\omega$"></SPAN> can be chosen
using some estimate of the spectral radius of <SPAN CLASS="MATH"><IMG
ALT="$\omega$"> can be chosen
using some estimate of the spectral radius of <IMG
WIDTH="50" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img85.png"
ALT="$D^{-1}A$"></SPAN> [<A
ALT="$D^{-1}A$"> [<A
HREF="node24.html#BREZINA_VANEK">1</A>].
<P>
<DIV CLASS="navigation"><HR>
<HR>
<!--Navigation Panel-->
<A NAME="tex2html204"
HREF="node13.html">
@@ -323,11 +318,11 @@ using some estimate of the spectral radius of <SPAN CLASS="MATH"><IMG
<B> Previous:</B> <A NAME="tex2html197"
HREF="node11.html">Multi-level Schwarz Preconditioners</A>
&nbsp; <B> <A NAME="tex2html203"
HREF="node2.html">Contents</A></B> </DIV>
HREF="node2.html">Contents</A></B>
<!--End of Navigation Panel-->
<ADDRESS>
Salvatore Filippone
2008-07-23
2008-07-24
</ADDRESS>
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