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mld2p4-2:

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Salvatore Filippone
2016-02-29 14:15:38 +00:00
parent 9c35e30fea
commit 1201cdfad8
137 changed files with 10543 additions and 1524 deletions
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docs/html/node12.html
+21 -22
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@@ -1,6 +1,6 @@
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<!--Converted with LaTeX2HTML 2008 (1.71)
<!--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:
@@ -13,7 +13,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
<META NAME="Generator" CONTENT="LaTeX2HTML v2008">
<META NAME="Generator" CONTENT="LaTeX2HTML v2012">
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
<LINK REL="STYLESHEET" HREF="userhtml.css">
@@ -91,7 +91,7 @@ and <!-- MATH
WIDTH="162" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img7.png"
ALT="$E=\{(i,j) : a_{ij} \neq 0\}$"> are the vertex set and the edge set of <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img8.png"
ALT="$G$">,
respectively. Two vertices are called adjacent if there is an edge connecting
@@ -99,15 +99,15 @@ them. For any integer <IMG
WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img9.png"
ALT="$\delta &gt; 0$">, a <IMG
WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img10.png"
ALT="$\delta$">-overlap
partition of <IMG
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
WIDTH="23" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img11.png"
ALT="$W$"> can be defined recursively as follows.
Given a 0-overlap (or non-overlapping) partition of <IMG
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
WIDTH="23" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img11.png"
ALT="$W$">,
i.e. a set of <IMG
@@ -127,11 +127,11 @@ i.e. a set of <IMG
WIDTH="108" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img14.png"
ALT="$\cup_{i=1}^m W_i^0 = W$">, a <IMG
WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img10.png"
ALT="$\delta$">-overlap
partition of <IMG
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
WIDTH="23" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img11.png"
ALT="$W$"> is obtained by considering the sets
<!-- MATH
@@ -193,7 +193,7 @@ prolongation operator from <!-- MATH
WIDTH="33" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
SRC="img24.png"
ALT="$\Re^{n_i^\delta}$"> to <IMG
WIDTH="26" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
WIDTH="26" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$\Re^n$">.
The matrix <!-- MATH
@@ -225,7 +225,7 @@ M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
-->
<IMG
WIDTH="206" HEIGHT="58" BORDER="0"
WIDTH="207" HEIGHT="58" BORDER="0"
SRC="img28.png"
ALT="\begin{displaymath}
M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
@@ -341,7 +341,7 @@ M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
-->
<IMG
WIDTH="216" HEIGHT="58" BORDER="0"
WIDTH="217" HEIGHT="58" BORDER="0"
SRC="img41.png"
ALT="\begin{displaymath}
M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
@@ -362,7 +362,7 @@ M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
-->
<IMG
WIDTH="218" HEIGHT="58" BORDER="0"
WIDTH="219" HEIGHT="58" BORDER="0"
SRC="img42.png"
ALT="\begin{displaymath}M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
(A_i^\delta)^{-1} \tilde{R}_i^0.
@@ -371,7 +371,7 @@ M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
<BR CLEAR="ALL">
<P></P>
We note that for <IMG
WIDTH="45" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
WIDTH="45" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img43.png"
ALT="$\delta=0$"> the three variants of the AS preconditioner are
all equal to the block-Jacobi preconditioner.
@@ -383,7 +383,7 @@ preconditioned iterative solvers deteriorates as the number <IMG
SRC="img12.png"
ALT="$m$"> of partitions
of <IMG
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
WIDTH="23" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img11.png"
ALT="$W$"> increases [<A
HREF="node25.html#dd1_94">7</A>,<A
@@ -450,7 +450,7 @@ M_{C}^{-1}= R_C^T A_C^{-1} R_C,
-->
<IMG
WIDTH="144" HEIGHT="32" BORDER="0"
WIDTH="145" HEIGHT="32" BORDER="0"
SRC="img50.png"
ALT="\begin{displaymath}
M_{C}^{-1}= R_C^T A_C^{-1} R_C,
@@ -566,7 +566,7 @@ M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
-->
<IMG
WIDTH="308" HEIGHT="34" BORDER="0"
WIDTH="309" HEIGHT="34" BORDER="0"
SRC="img57.png"
ALT="\begin{displaymath}
M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
@@ -609,7 +609,7 @@ M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
-->
<IMG
WIDTH="316" HEIGHT="34" BORDER="0"
WIDTH="318" HEIGHT="34" BORDER="0"
SRC="img59.png"
ALT="\begin{displaymath}
M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
@@ -662,12 +662,12 @@ computation of <IMG
ALT="$w=M^{-1}v$">, is reported, for
example, in Figure&nbsp;<A HREF="#fig:mlhpost_alg">1</A>. Here the number of levels
is denoted by <IMG
WIDTH="38" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
WIDTH="38" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img62.png"
ALT="$nlev$"> and the levels are numbered in increasing order starting
from the finest one, i.e. the finest level is level 1; the coarse matrix
and the corresponding basic preconditioner at each level <IMG
WIDTH="10" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
WIDTH="10" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img63.png"
ALT="$l$"> are denoted by <IMG
WIDTH="23" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
@@ -695,8 +695,7 @@ Application of the multi-level hybrid post-smoothed preconditioner.</CAPTION>
$\framebox{
\begin{minipage}{.85\textwidth} {\small
\begin{tabbing}
\quad \=\quad \=\quad \=\quad \\[-1mm]
$v_1 = v$; \\[2mm]
\quad \=\quad \=\quad \=\quad \\[-1mm]$v_1 = v$; \\[2mm]
\textbf{for $l=2, nlev$\ do}\\[1mm]
\> ! transfer $v_{l-1}$\ to the next coarser level\\
\> $v_l = R_lv_{l-1}$\ \\[1mm]
@@ -727,7 +726,7 @@ $w = y_1$;
\begin{tabbing}
\quad \=\quad...
...= y_l+r_l$\\
\textbf{endfor} \\ [1mm]
\textbf{endfor} \ [1mm]
$w = y_1$;
\end{tabbing}}
\end{minipage}}">