mld2p4-2:

Version number string &  docs.
This commit is contained in:
Salvatore Filippone
2011-03-25 16:30:04 +00:00
parent 1a43c78bf1
commit 0288cf909a
149 changed files with 1319 additions and 1323 deletions
+150 -158
View File
@@ -1,6 +1,6 @@
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
<!--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:
@@ -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 v2002-2-1">
<META NAME="Generator" CONTENT="LaTeX2HTML v2008">
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
<LINK REL="STYLESHEET" HREF="userhtml.css">
@@ -28,20 +28,16 @@ original version by: Nikos Drakos, CBLU, University of Leeds
<!--Navigation Panel-->
<A NAME="tex2html209"
HREF="node13.html">
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<B> Next:</B> <A NAME="tex2html210"
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@@ -72,12 +68,12 @@ where <!-- MATH
-->
<IMG
WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img3.png"
SRC="img4.png"
ALT="$A=(a_{ij}) \in \Re^{n \times n}$"> is a
nonsingular sparse matrix with a symmetric nonzero pattern,
let <IMG
WIDTH="93" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img4.png"
WIDTH="92" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img5.png"
ALT="$G=(W,E)$"> be the adjacency graph of <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img2.png"
@@ -85,136 +81,136 @@ let <IMG
$W=\{1, 2, \ldots, n\}$
-->
<IMG
WIDTH="138" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img5.png"
WIDTH="139" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img6.png"
ALT="$W=\{1, 2, \ldots, n\}$">
and <!-- MATH
$E=\{(i,j) : a_{ij} \neq 0\}$
-->
<IMG
WIDTH="162" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img6.png"
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"
SRC="img7.png"
SRC="img8.png"
ALT="$G$">,
respectively. Two vertices are called adjacent if there is an edge connecting
them. For any integer <IMG
WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img8.png"
SRC="img9.png"
ALT="$\delta &gt; 0$">, a <IMG
WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
SRC="img10.png"
ALT="$\delta$">-overlap
partition of <IMG
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img10.png"
WIDTH="23" HEIGHT="15" 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="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img10.png"
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img11.png"
ALT="$W$">,
i.e. a set of <IMG
WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img11.png"
WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img12.png"
ALT="$m$"> disjoint nonempty sets <!-- MATH
$W_i^0 \subset W$
-->
<IMG
WIDTH="73" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img12.png"
SRC="img13.png"
ALT="$W_i^0 \subset W$"> such that
<!-- MATH
$\cup_{i=1}^m W_i^0 = W$
-->
<IMG
WIDTH="107" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img13.png"
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"
SRC="img9.png"
SRC="img10.png"
ALT="$\delta$">-overlap
partition of <IMG
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img10.png"
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img11.png"
ALT="$W$"> is obtained by considering the sets
<!-- MATH
$W_i^\delta \supset W_i^{\delta-1}$
-->
<IMG
WIDTH="97" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img14.png"
SRC="img15.png"
ALT="$W_i^\delta \supset W_i^{\delta-1}$"> obtained by including the vertices that
are adjacent to any vertex in <!-- MATH
$W_i^{\delta-1}$
-->
<IMG
WIDTH="48" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img15.png"
SRC="img16.png"
ALT="$W_i^{\delta-1}$">.
<P>
Let <IMG
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img16.png"
ALT="$n_i^\delta$"> be the size of <IMG
WIDTH="30" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img17.png"
ALT="$n_i^\delta$"> be the size of <IMG
WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img18.png"
ALT="$W_i^\delta$"> and <!-- MATH
$R_i^{\delta} \in
\Re^{n_i^\delta \times n}$
-->
<IMG
WIDTH="93" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
SRC="img18.png"
WIDTH="93" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
SRC="img19.png"
ALT="$R_i^{\delta} \in
\Re^{n_i^\delta \times n}$"> the restriction operator that maps
a vector <IMG
WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img19.png"
WIDTH="57" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img20.png"
ALT="$v \in \Re^n$"> onto the vector <!-- MATH
$v_i^{\delta} \in \Re^{n_i^\delta}$
-->
<IMG
WIDTH="70" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
SRC="img20.png"
WIDTH="70" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
SRC="img21.png"
ALT="$v_i^{\delta} \in \Re^{n_i^\delta}$">
containing the components of <IMG
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img22.png"
ALT="$v$"> corresponding to the vertices in
<IMG
WIDTH="30" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img17.png"
WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img18.png"
ALT="$W_i^\delta$">. The transpose of <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img22.png"
SRC="img23.png"
ALT="$R_i^{\delta}$"> is a
prolongation operator from <!-- MATH
$\Re^{n_i^\delta}$
-->
<IMG
WIDTH="32" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
SRC="img23.png"
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"
SRC="img24.png"
SRC="img25.png"
ALT="$\Re^n$">.
The matrix <!-- MATH
$A_i^\delta=R_i^\delta A (R_i^\delta)^T \in
\Re^{n_i^\delta \times n_i^\delta}$
-->
<IMG
WIDTH="201" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
SRC="img25.png"
WIDTH="201" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
SRC="img26.png"
ALT="$A_i^\delta=R_i^\delta A (R_i^\delta)^T \in
\Re^{n_i^\delta \times n_i^\delta}$"> can be considered
as a restriction of <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img2.png"
ALT="$A$"> corresponding to the set <IMG
WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img26.png"
WIDTH="30" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img27.png"
ALT="$W_i^{\delta}$">.
<P>
@@ -230,7 +226,7 @@ M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
<IMG
WIDTH="206" HEIGHT="58" BORDER="0"
SRC="img27.png"
SRC="img28.png"
ALT="\begin{displaymath}
M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
(A_i^\delta)^{-1} R_i^{\delta},
@@ -240,27 +236,27 @@ M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
<P></P>
where <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img28.png"
SRC="img29.png"
ALT="$A_i^\delta$"> is assumed to be nonsingular. Its application
to a vector <IMG
WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img19.png"
WIDTH="57" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img20.png"
ALT="$v \in \Re^n$"> within a Krylov solver requires the following
three steps:
<OL>
<LI>restriction of <IMG
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img22.png"
ALT="$v$"> as <!-- MATH
$v_i = R_i^{\delta} v$
-->
<IMG
WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img29.png"
ALT="$v_i = R_i^{\delta} v$">, <IMG
WIDTH="97" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
SRC="img30.png"
ALT="$v_i = R_i^{\delta} v$">, <IMG
WIDTH="96" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
SRC="img31.png"
ALT="$i=1,\ldots,m$">;
</LI>
<LI>solution of the linear systems <!-- MATH
@@ -268,36 +264,36 @@ three steps:
-->
<IMG
WIDTH="80" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img31.png"
SRC="img32.png"
ALT="$A_i^\delta w_i = v_i$">,
<IMG
WIDTH="97" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
SRC="img30.png"
WIDTH="96" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
SRC="img31.png"
ALT="$i=1,\ldots,m$">;
</LI>
<LI>prolongation and sum of the <IMG
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img32.png"
WIDTH="22" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img33.png"
ALT="$w_i$">'s, i.e. <!-- MATH
$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$
-->
<IMG
WIDTH="144" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img33.png"
WIDTH="145" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img34.png"
ALT="$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$">.
</LI>
</OL>
Note that the linear systems at step 2 are usually solved approximately,
e.g. using incomplete LU factorizations such as ILU(<IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img34.png"
SRC="img35.png"
ALT="$p$">), MILU(<IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img34.png"
SRC="img35.png"
ALT="$p$">) and
ILU(<IMG
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img35.png"
SRC="img36.png"
ALT="$p,t$">) [<A
HREF="node25.html#Saad_book">20</A>, Chapter 10].
@@ -309,8 +305,8 @@ time on parallel distributed-memory computers is the so-called <I>Restricted AS
HREF="node25.html#CAI_SARKIS">5</A>,<A
HREF="node25.html#EFSTATHIOU">14</A>]. It
is obtained by zeroing the components of <IMG
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img32.png"
WIDTH="22" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img33.png"
ALT="$w_i$"> corresponding to the
overlapping vertices when applying the prolongation. Therefore,
RAS differs from classical AS by the prolongation operators,
@@ -318,22 +314,22 @@ which are substituted by <!-- MATH
$(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$
-->
<IMG
WIDTH="118" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
SRC="img36.png"
WIDTH="117" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
SRC="img37.png"
ALT="$(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$">,
where <IMG
WIDTH="25" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
SRC="img37.png"
WIDTH="26" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
SRC="img38.png"
ALT="$\tilde{R}_i^0$"> is obtained by zeroing the rows of <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img38.png"
SRC="img39.png"
ALT="$R_i^\delta$">
corresponding to the vertices in <!-- MATH
$W_i^\delta \backslash W_i^0$
-->
<IMG
WIDTH="66" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img39.png"
SRC="img40.png"
ALT="$W_i^\delta \backslash W_i^0$">:
<BR><P></P>
<DIV ALIGN="CENTER">
@@ -346,7 +342,7 @@ M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
<IMG
WIDTH="216" HEIGHT="58" BORDER="0"
SRC="img40.png"
SRC="img41.png"
ALT="\begin{displaymath}
M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
(A_i^\delta)^{-1} R_i^{\delta}.
@@ -367,7 +363,7 @@ M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
<IMG
WIDTH="218" HEIGHT="58" BORDER="0"
SRC="img41.png"
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.
\end{displaymath}">
@@ -376,19 +372,19 @@ M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
<P></P>
We note that for <IMG
WIDTH="45" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img42.png"
SRC="img43.png"
ALT="$\delta=0$"> the three variants of the AS preconditioner are
all equal to the block-Jacobi preconditioner.
<P>
As already observed, the convergence rate of the one-level Schwarz
preconditioned iterative solvers deteriorates as the number <IMG
WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img11.png"
WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img12.png"
ALT="$m$"> of partitions
of <IMG
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img10.png"
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img11.png"
ALT="$W$"> increases [<A
HREF="node25.html#dd1_94">7</A>,<A
HREF="node25.html#dd2_96">21</A>]. To reduce the dependency
@@ -396,32 +392,32 @@ of the number of iterations on the degree of parallelism we may
introduce a global coupling among the overlapping partitions by defining
a coarse-space approximation <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img43.png"
SRC="img44.png"
ALT="$A_C$"> of the matrix <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img2.png"
ALT="$A$">.
In a pure algebraic setting, <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img43.png"
SRC="img44.png"
ALT="$A_C$"> is usually built with
the Galerkin approach. Given a set <IMG
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img44.png"
SRC="img45.png"
ALT="$W_C$"> of <I>coarse vertices</I>,
with size <IMG
WIDTH="26" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img45.png"
SRC="img46.png"
ALT="$n_C$">, and a suitable restriction operator
<!-- MATH
$R_C \in \Re^{n_C \times n}$
-->
<IMG
WIDTH="101" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img46.png"
WIDTH="100" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img47.png"
ALT="$R_C \in \Re^{n_C \times n}$">, <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img43.png"
SRC="img44.png"
ALT="$A_C$"> is defined as
<BR><P></P>
<DIV ALIGN="CENTER">
@@ -433,7 +429,7 @@ A_C=R_C A R_C^T
<IMG
WIDTH="109" HEIGHT="31" BORDER="0"
SRC="img47.png"
SRC="img48.png"
ALT="\begin{displaymath}
A_C=R_C A R_C^T
\end{displaymath}">
@@ -443,7 +439,7 @@ A_C=R_C A R_C^T
and the coarse-level correction matrix to be combined with a generic
one-level AS preconditioner <IMG
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img48.png"
SRC="img49.png"
ALT="$M_{1L}$"> is obtained as
<BR><P></P>
<DIV ALIGN="CENTER">
@@ -455,7 +451,7 @@ M_{C}^{-1}= R_C^T A_C^{-1} R_C,
<IMG
WIDTH="144" HEIGHT="32" BORDER="0"
SRC="img49.png"
SRC="img50.png"
ALT="\begin{displaymath}
M_{C}^{-1}= R_C^T A_C^{-1} R_C,
\end{displaymath}">
@@ -464,28 +460,28 @@ M_{C}^{-1}= R_C^T A_C^{-1} R_C,
<P></P>
where <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img43.png"
SRC="img44.png"
ALT="$A_C$"> is assumed to be nonsingular. The application of <IMG
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img50.png"
WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$M_{C}^{-1}$">
to a vector <IMG
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img22.png"
ALT="$v$"> corresponds to a restriction, a solution and
a prolongation step; the solution step, involving the matrix <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img43.png"
SRC="img44.png"
ALT="$A_C$">,
may be carried out also approximately.
<P>
The combination of <IMG
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img52.png"
ALT="$M_{C}$"> and <IMG
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img48.png"
SRC="img49.png"
ALT="$M_{1L}$"> may be
performed in either an additive or a multiplicative framework.
In the former case, the <I>two-level additive</I> Schwarz preconditioner
@@ -499,8 +495,8 @@ M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
-->
<IMG
WIDTH="165" HEIGHT="32" BORDER="0"
SRC="img52.png"
WIDTH="166" HEIGHT="32" BORDER="0"
SRC="img53.png"
ALT="\begin{displaymath}
M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
\end{displaymath}">
@@ -509,33 +505,33 @@ M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
<P></P>
Applying <IMG
WIDTH="59" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img53.png"
SRC="img54.png"
ALT="$M_{2L-A}^{-1}$"> to a vector <IMG
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img22.png"
ALT="$v$"> within a Krylov solver
corresponds to applying <IMG
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img50.png"
WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$M_{C}^{-1}$">
and <IMG
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img54.png"
WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img55.png"
ALT="$M_{1L}^{-1}$"> to <IMG
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img22.png"
ALT="$v$"> independently and then summing up
the results.
<P>
In the multiplicative case, the combination can be
performed by first applying the smoother <IMG
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img54.png"
WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img55.png"
ALT="$M_{1L}^{-1}$"> and then
the coarse-level correction operator <IMG
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img50.png"
WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$M_{C}^{-1}$">:
<BR><P></P>
<DIV ALIGN="CENTER">
@@ -550,7 +546,7 @@ z = w + M_{C}^{-1} (v-Aw);
<IMG
WIDTH="177" HEIGHT="51" BORDER="0"
SRC="img55.png"
SRC="img56.png"
ALT="\begin{displaymath}
\begin{array}{l}
w = M_{1L}^{-1} v, \\
@@ -570,8 +566,8 @@ M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
-->
<IMG
WIDTH="308" HEIGHT="33" BORDER="0"
SRC="img56.png"
WIDTH="308" 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}.
\end{displaymath}">
@@ -593,7 +589,7 @@ z = w + M_{1L}^{-1} (v-Aw) ,
<IMG
WIDTH="177" HEIGHT="51" BORDER="0"
SRC="img57.png"
SRC="img58.png"
ALT="\begin{displaymath}
\begin{array}{l}
w = M_{C}^{-1} v , \\
@@ -613,8 +609,8 @@ M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
-->
<IMG
WIDTH="317" HEIGHT="33" BORDER="0"
SRC="img58.png"
WIDTH="316" 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}.
\end{displaymath}">
@@ -628,10 +624,10 @@ preconditioner is symmetric if <IMG
SRC="img2.png"
ALT="$A$">, <IMG
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img48.png"
SRC="img49.png"
ALT="$M_{1L}$"> and <IMG
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img52.png"
ALT="$M_{C}$"> are symmetric.
<P>
@@ -655,41 +651,41 @@ referred to [<A
The algorithm for the application of a multi-level hybrid
post-smoothed preconditioner <IMG
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img59.png"
SRC="img60.png"
ALT="$M$"> to a vector <IMG
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img22.png"
ALT="$v$">, i.e. for the
computation of <IMG
WIDTH="87" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img60.png"
WIDTH="86" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img61.png"
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="37" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img61.png"
WIDTH="38" HEIGHT="15" 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"
SRC="img62.png"
ALT="$l$"> are denoted by <IMG
WIDTH="22" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img63.png"
ALT="$l$"> are denoted by <IMG
WIDTH="23" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img64.png"
ALT="$A_l$"> and
<IMG
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img64.png"
ALT="$M_l$">, respectively, with <IMG
WIDTH="61" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img65.png"
ALT="$M_l$">, respectively, with <IMG
WIDTH="62" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img66.png"
ALT="$A_1=A$">, while the related restriction operator is
denoted by <IMG
WIDTH="22" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img66.png"
WIDTH="23" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img67.png"
ALT="$R_l$">.
<DIV ALIGN="CENTER"><A NAME="fig:mlhpost_alg"></A><A NAME="508"></A>
<DIV ALIGN="CENTER"><A NAME="fig:mlhpost_alg"></A><A NAME="512"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
Application of the multi-level hybrid post-smoothed preconditioner.</CAPTION>
@@ -725,13 +721,13 @@ $w = y_1$;
-->
<IMG
WIDTH="430" HEIGHT="435" ALIGN="BOTTOM" BORDER="0"
SRC="img67.png"
SRC="img68.png"
ALT="\framebox{
\begin{minipage}{.85\textwidth} {\small
\begin{tabbing}
\quad \=\quad...
...= y_l+r_l$\\
\textbf{endfor} \ [1mm]
\textbf{endfor} \\ [1mm]
$w = y_1$;
\end{tabbing}}
\end{minipage}}">
@@ -745,20 +741,16 @@ $w = y_1$;
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