Further doc fixes.
This commit is contained in:
Salvatore Filippone
2008-07-24 06:05:46 +00:00
parent 4dda0494a7
commit 78805bf653
130 changed files with 10569 additions and 7036 deletions
+171 -173
View File
@@ -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
@@ -25,8 +25,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds
</HEAD>
<BODY >
<DIV CLASS="navigation"><!--Navigation Panel-->
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<A NAME="tex2html194"
HREF="node12.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
@@ -53,7 +52,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds
&nbsp; <B> <A NAME="tex2html193"
HREF="node2.html">Contents</A></B>
<BR>
<BR></DIV>
<BR>
<!--End of Navigation Panel-->
<H2><A NAME="SECTION00061000000000000000"></A><A NAME="sec:multilevel"></A>
@@ -71,157 +70,157 @@ Given the linear system ,
where <!-- MATH
$A=(a_{ij}) \in \Re^{n \times n}$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img3.png"
ALT="$A=(a_{ij}) \in \Re^{n \times n}$"></SPAN> is a
ALT="$A=(a_{ij}) \in \Re^{n \times n}$"> is a
nonsingular sparse matrix with a symmetric nonzero pattern,
let <SPAN CLASS="MATH"><IMG
let <IMG
WIDTH="93" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img4.png"
ALT="$G=(W,E)$"></SPAN> be the adjacency graph of <SPAN CLASS="MATH"><IMG
ALT="$G=(W,E)$"> be the adjacency graph of <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img2.png"
ALT="$A$"></SPAN>, where <!-- MATH
ALT="$A$">, where <!-- MATH
$W=\{1, 2, \ldots, n\}$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="138" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img5.png"
ALT="$W=\{1, 2, \ldots, n\}$"></SPAN>
ALT="$W=\{1, 2, \ldots, n\}$">
and <!-- MATH
$E=\{(i,j) : a_{ij} \neq 0\}$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="162" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img6.png"
ALT="$E=\{(i,j) : a_{ij} \neq 0\}$"></SPAN> are the vertex set and the edge set of <SPAN CLASS="MATH"><IMG
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"
ALT="$G$"></SPAN>,
ALT="$G$">,
respectively. Two vertices are called adjacent if there is an edge connecting
them. For any integer <SPAN CLASS="MATH"><IMG
them. For any integer <IMG
WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img8.png"
ALT="$\delta &gt; 0$"></SPAN>, a <SPAN CLASS="MATH"><IMG
ALT="$\delta &gt; 0$">, a <IMG
WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\delta$"></SPAN>-overlap
partition of <SPAN CLASS="MATH"><IMG
ALT="$\delta$">-overlap
partition of <IMG
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img10.png"
ALT="$W$"></SPAN> can be defined recursively as follows.
Given a 0-overlap (or non-overlapping) partition of <SPAN CLASS="MATH"><IMG
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"
ALT="$W$"></SPAN>,
i.e. a set of <SPAN CLASS="MATH"><IMG
ALT="$W$">,
i.e. a set of <IMG
WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img11.png"
ALT="$m$"></SPAN> disjoint nonempty sets <!-- MATH
ALT="$m$"> disjoint nonempty sets <!-- MATH
$W_i^0 \subset W$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="73" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img12.png"
ALT="$W_i^0 \subset W$"></SPAN> such that
ALT="$W_i^0 \subset W$"> such that
<!-- MATH
$\cup_{i=1}^m W_i^0 = W$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="107" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img13.png"
ALT="$\cup_{i=1}^m W_i^0 = W$"></SPAN>, a <SPAN CLASS="MATH"><IMG
ALT="$\cup_{i=1}^m W_i^0 = W$">, a <IMG
WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\delta$"></SPAN>-overlap
partition of <SPAN CLASS="MATH"><IMG
ALT="$\delta$">-overlap
partition of <IMG
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img10.png"
ALT="$W$"></SPAN> is obtained by considering the sets
ALT="$W$"> is obtained by considering the sets
<!-- MATH
$W_i^\delta \supset W_i^{\delta-1}$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="97" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img14.png"
ALT="$W_i^\delta \supset W_i^{\delta-1}$"></SPAN> obtained by including the vertices that
ALT="$W_i^\delta \supset W_i^{\delta-1}$"> obtained by including the vertices that
are adjacent to any vertex in <!-- MATH
$W_i^{\delta-1}$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="48" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img15.png"
ALT="$W_i^{\delta-1}$"></SPAN>.
ALT="$W_i^{\delta-1}$">.
<P>
Let <SPAN CLASS="MATH"><IMG
Let <IMG
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img16.png"
ALT="$n_i^\delta$"></SPAN> be the size of <SPAN CLASS="MATH"><IMG
ALT="$n_i^\delta$"> be the size of <IMG
WIDTH="30" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img17.png"
ALT="$W_i^\delta$"></SPAN> and <!-- MATH
ALT="$W_i^\delta$"> and <!-- MATH
$R_i^{\delta} \in
\Re^{n_i^\delta \times n}$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="93" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
SRC="img18.png"
ALT="$R_i^{\delta} \in
\Re^{n_i^\delta \times n}$"></SPAN> the restriction operator that maps
a vector <SPAN CLASS="MATH"><IMG
\Re^{n_i^\delta \times n}$"> the restriction operator that maps
a vector <IMG
WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img19.png"
ALT="$v \in \Re^n$"></SPAN> onto the vector <!-- MATH
ALT="$v \in \Re^n$"> onto the vector <!-- MATH
$v_i^{\delta} \in \Re^{n_i^\delta}$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="70" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
SRC="img20.png"
ALT="$v_i^{\delta} \in \Re^{n_i^\delta}$"></SPAN>
containing the components of <SPAN CLASS="MATH"><IMG
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"
ALT="$v$"></SPAN> corresponding to the vertices in
<SPAN CLASS="MATH"><IMG
ALT="$v$"> corresponding to the vertices in
<IMG
WIDTH="30" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img17.png"
ALT="$W_i^\delta$"></SPAN>. The transpose of <SPAN CLASS="MATH"><IMG
ALT="$W_i^\delta$">. The transpose of <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img22.png"
ALT="$R_i^{\delta}$"></SPAN> is a
ALT="$R_i^{\delta}$"> is a
prolongation operator from <!-- MATH
$\Re^{n_i^\delta}$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="32" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
SRC="img23.png"
ALT="$\Re^{n_i^\delta}$"></SPAN> to <SPAN CLASS="MATH"><IMG
ALT="$\Re^{n_i^\delta}$"> to <IMG
WIDTH="26" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img24.png"
ALT="$\Re^n$"></SPAN>.
ALT="$\Re^n$">.
The matrix <!-- MATH
$A_i^\delta=R_i^\delta A (R_i^\delta)^T \in
\Re^{n_i^\delta \times n_i^\delta}$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="201" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
SRC="img25.png"
ALT="$A_i^\delta=R_i^\delta A (R_i^\delta)^T \in
\Re^{n_i^\delta \times n_i^\delta}$"></SPAN> can be considered
as a restriction of <SPAN CLASS="MATH"><IMG
\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$"></SPAN> corresponding to the set <SPAN CLASS="MATH"><IMG
ALT="$A$"> corresponding to the set <IMG
WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img26.png"
ALT="$W_i^{\delta}$"></SPAN>.
ALT="$W_i^{\delta}$">.
<P>
The <SPAN CLASS="textit">classical one-level AS</SPAN> preconditioner is defined by
The <I>classical one-level AS</I> preconditioner is defined by
<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
@@ -239,105 +238,105 @@ M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
</DIV>
<BR CLEAR="ALL">
<P></P>
where <SPAN CLASS="MATH"><IMG
where <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img28.png"
ALT="$A_i^\delta$"></SPAN> is assumed to be nonsingular. Its application
to a vector <SPAN CLASS="MATH"><IMG
ALT="$A_i^\delta$"> is assumed to be nonsingular. Its application
to a vector <IMG
WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img19.png"
ALT="$v \in \Re^n$"></SPAN> within a Krylov solver requires the following
ALT="$v \in \Re^n$"> within a Krylov solver requires the following
three steps:
<OL>
<LI>restriction of <SPAN CLASS="MATH"><IMG
<LI>restriction of <IMG
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
ALT="$v$"></SPAN> as <!-- MATH
ALT="$v$"> as <!-- MATH
$v_i = R_i^{\delta} v$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img29.png"
ALT="$v_i = R_i^{\delta} v$"></SPAN>, <SPAN CLASS="MATH"><IMG
ALT="$v_i = R_i^{\delta} v$">, <IMG
WIDTH="97" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
SRC="img30.png"
ALT="$i=1,\ldots,m$"></SPAN>;
ALT="$i=1,\ldots,m$">;
</LI>
<LI>solution of the linear systems <!-- MATH
$A_i^\delta w_i = v_i$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="80" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img31.png"
ALT="$A_i^\delta w_i = v_i$"></SPAN>,
<SPAN CLASS="MATH"><IMG
ALT="$A_i^\delta w_i = v_i$">,
<IMG
WIDTH="97" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
SRC="img30.png"
ALT="$i=1,\ldots,m$"></SPAN>;
ALT="$i=1,\ldots,m$">;
</LI>
<LI>prolongation and sum of the <SPAN CLASS="MATH"><IMG
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
<LI>prolongation and sum of the <IMG
WIDTH="22" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img32.png"
ALT="$w_i$"></SPAN>'s, i.e. <!-- MATH
ALT="$w_i$">'s, i.e. <!-- MATH
$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="144" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img33.png"
ALT="$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$"></SPAN>.
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(<SPAN CLASS="MATH"><IMG
e.g. using incomplete LU factorizations such as ILU(<IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img34.png"
ALT="$p$"></SPAN>), MILU(<SPAN CLASS="MATH"><IMG
ALT="$p$">), MILU(<IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img34.png"
ALT="$p$"></SPAN>) and
ILU(<SPAN CLASS="MATH"><IMG
ALT="$p$">) and
ILU(<IMG
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img35.png"
ALT="$p,t$"></SPAN>) [<A
ALT="$p,t$">) [<A
HREF="node24.html#Saad_book">19</A>, Chapter 10].
<P>
A variant of the classical AS preconditioner that outperforms it
in terms of convergence rate and of computation and communication
time on parallel distributed-memory computers is the so-called <SPAN CLASS="textit">Restricted AS
(RAS)</SPAN> preconditioner&nbsp;[<A
time on parallel distributed-memory computers is the so-called <I>Restricted AS
(RAS)</I> preconditioner&nbsp;[<A
HREF="node24.html#CAI_SARKIS">5</A>,<A
HREF="node24.html#EFSTATHIOU">13</A>]. It
is obtained by zeroing the components of <SPAN CLASS="MATH"><IMG
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
is obtained by zeroing the components of <IMG
WIDTH="22" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img32.png"
ALT="$w_i$"></SPAN> corresponding to the
ALT="$w_i$"> corresponding to the
overlapping vertices when applying the prolongation. Therefore,
RAS differs from classical AS by the prolongation operators,
which are substituted by <!-- MATH
$(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="118" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
SRC="img36.png"
ALT="$(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$"></SPAN>,
where <SPAN CLASS="MATH"><IMG
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"
ALT="$\tilde{R}_i^0$"></SPAN> is obtained by zeroing the rows of <SPAN CLASS="MATH"><IMG
ALT="$\tilde{R}_i^0$"> is obtained by zeroing the rows of <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img38.png"
ALT="$R_i^\delta$"></SPAN>
ALT="$R_i^\delta$">
corresponding to the vertices in <!-- MATH
$W_i^\delta \backslash W_i^0$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="66" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img39.png"
ALT="$W_i^\delta \backslash W_i^0$"></SPAN>:
ALT="$W_i^\delta \backslash W_i^0$">:
<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
@@ -355,10 +354,10 @@ M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
</DIV>
<BR CLEAR="ALL">
<P></P>
Analogously, the AS variant called <SPAN CLASS="textit">AS with Harmonic extension (ASH)</SPAN>
Analogously, the AS variant called <I>AS with Harmonic extension (ASH)</I>
is defined by
<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
@@ -375,57 +374,57 @@ M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
</DIV>
<BR CLEAR="ALL">
<P></P>
We note that for <SPAN CLASS="MATH"><IMG
We note that for <IMG
WIDTH="45" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img42.png"
ALT="$\delta=0$"></SPAN> the three variants of the AS preconditioner are
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 <SPAN CLASS="MATH"><IMG
preconditioned iterative solvers deteriorates as the number <IMG
WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img11.png"
ALT="$m$"></SPAN> of partitions
of <SPAN CLASS="MATH"><IMG
ALT="$m$"> of partitions
of <IMG
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img10.png"
ALT="$W$"></SPAN> increases [<A
ALT="$W$"> increases [<A
HREF="node24.html#dd1_94">7</A>,<A
HREF="node24.html#dd2_96">20</A>]. To reduce the dependency
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 <SPAN CLASS="MATH"><IMG
a coarse-space approximation <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img43.png"
ALT="$A_C$"></SPAN> of the matrix <SPAN CLASS="MATH"><IMG
ALT="$A_C$"> of the matrix <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img2.png"
ALT="$A$"></SPAN>.
In a pure algebraic setting, <SPAN CLASS="MATH"><IMG
ALT="$A$">.
In a pure algebraic setting, <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img43.png"
ALT="$A_C$"></SPAN> is usually built with
a Galerkin approach. Given a set <SPAN CLASS="MATH"><IMG
ALT="$A_C$"> is usually built with
a Galerkin approach. Given a set <IMG
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img44.png"
ALT="$W_C$"></SPAN> of <SPAN CLASS="textit">coarse vertices</SPAN>,
with size <SPAN CLASS="MATH"><IMG
ALT="$W_C$"> of <I>coarse vertices</I>,
with size <IMG
WIDTH="26" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img45.png"
ALT="$n_C$"></SPAN>, and a suitable restriction operator
ALT="$n_C$">, and a suitable restriction operator
<!-- MATH
$R_C \in \Re^{n_C \times n}$
-->
<SPAN CLASS="MATH"><IMG
<IMG
WIDTH="101" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img46.png"
ALT="$R_C \in \Re^{n_C \times n}$"></SPAN>, <SPAN CLASS="MATH"><IMG
ALT="$R_C \in \Re^{n_C \times n}$">, <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img43.png"
ALT="$A_C$"></SPAN> is defined as
ALT="$A_C$"> is defined as
<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A_C=R_C A R_C^T
@@ -442,12 +441,12 @@ A_C=R_C A R_C^T
<BR CLEAR="ALL">
<P></P>
and the coarse-level correction matrix to be combined with a generic
one-level AS preconditioner <SPAN CLASS="MATH"><IMG
one-level AS preconditioner <IMG
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img48.png"
ALT="$M_{1L}$"></SPAN> is obtained as
ALT="$M_{1L}$"> is obtained as
<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
M_{C}^{-1}= R_C^T A_C^{-1} R_C,
@@ -463,36 +462,36 @@ M_{C}^{-1}= R_C^T A_C^{-1} R_C,
</DIV>
<BR CLEAR="ALL">
<P></P>
where <SPAN CLASS="MATH"><IMG
where <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img43.png"
ALT="$A_C$"></SPAN> is assumed to be nonsingular. The application of <SPAN CLASS="MATH"><IMG
ALT="$A_C$"> is assumed to be nonsingular. The application of <IMG
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img50.png"
ALT="$M_{C}^{-1}$"></SPAN>
to a vector <SPAN CLASS="MATH"><IMG
ALT="$M_{C}^{-1}$">
to a vector <IMG
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
ALT="$v$"></SPAN> corresponds to a restriction, a solution and
a prolongation step; the solution step, involving the matrix <SPAN CLASS="MATH"><IMG
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"
ALT="$A_C$"></SPAN>,
ALT="$A_C$">,
may be carried out also approximately.
<P>
The combination of <SPAN CLASS="MATH"><IMG
The combination of <IMG
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$M_{C}$"></SPAN> and <SPAN CLASS="MATH"><IMG
ALT="$M_{C}$"> and <IMG
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img48.png"
ALT="$M_{1L}$"></SPAN> may be
ALT="$M_{1L}$"> may be
performed in either an additive or a multiplicative framework.
In the former case, the <SPAN CLASS="textit">two-level additive</SPAN> Schwarz preconditioner
In the former case, the <I>two-level additive</I> Schwarz preconditioner
is obtained:
<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
@@ -500,7 +499,7 @@ M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
-->
<IMG
WIDTH="165" HEIGHT="32" BORDER="0"
WIDTH="166" HEIGHT="32" BORDER="0"
SRC="img52.png"
ALT="\begin{displaymath}
M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
@@ -508,38 +507,38 @@ M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
</DIV>
<BR CLEAR="ALL">
<P></P>
Applying <SPAN CLASS="MATH"><IMG
Applying <IMG
WIDTH="59" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img53.png"
ALT="$M_{2L-A}^{-1}$"></SPAN> to a vector <SPAN CLASS="MATH"><IMG
ALT="$M_{2L-A}^{-1}$"> to a vector <IMG
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
ALT="$v$"></SPAN> within a Krylov solver
corresponds to applying <SPAN CLASS="MATH"><IMG
ALT="$v$"> within a Krylov solver
corresponds to applying <IMG
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img50.png"
ALT="$M_{C}^{-1}$"></SPAN>
and <SPAN CLASS="MATH"><IMG
ALT="$M_{C}^{-1}$">
and <IMG
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img54.png"
ALT="$M_{1L}^{-1}$"></SPAN> to <SPAN CLASS="MATH"><IMG
ALT="$M_{1L}^{-1}$"> to <IMG
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
ALT="$v$"></SPAN> independently and then summing up
ALT="$v$"> independently and then summing up
the results.
<P>
In the multiplicative case, the combination can be
performed by first applying the smoother <SPAN CLASS="MATH"><IMG
performed by first applying the smoother <IMG
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img54.png"
ALT="$M_{1L}^{-1}$"></SPAN> and then
the coarse-level correction operator <SPAN CLASS="MATH"><IMG
ALT="$M_{1L}^{-1}$"> and then
the coarse-level correction operator <IMG
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img50.png"
ALT="$M_{C}^{-1}$"></SPAN>:
ALT="$M_{C}^{-1}$">:
<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\begin{array}{l}
@@ -560,10 +559,10 @@ z = w + M_{C}^{-1} (v-Aw);
</DIV>
<BR CLEAR="ALL">
<P></P>
this corresponds to the following <SPAN CLASS="textit">two-level hybrid pre-smoothed</SPAN>
this corresponds to the following <I>two-level hybrid pre-smoothed</I>
Schwarz preconditioner:
<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
@@ -582,7 +581,7 @@ M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
On the other hand, by applying the smoother after the coarse-level correction,
i.e. by computing
<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\begin{array}{l}
@@ -603,10 +602,10 @@ z = w + M_{1L}^{-1} (v-Aw) ,
</DIV>
<BR CLEAR="ALL">
<P></P>
the <SPAN CLASS="textit">two-level hybrid post-smoothed</SPAN>
the <I>two-level hybrid post-smoothed</I>
Schwarz preconditioner is obtained:
<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
@@ -614,7 +613,7 @@ M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
-->
<IMG
WIDTH="317" HEIGHT="33" BORDER="0"
WIDTH="316" HEIGHT="33" BORDER="0"
SRC="img58.png"
ALT="\begin{displaymath}
M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
@@ -624,16 +623,16 @@ M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
<P></P>
One more variant of two-level hybrid preconditioner is obtained by applying
the smoother before and after the coarse-level correction. In this case, the
preconditioner is symmetric if <SPAN CLASS="MATH"><IMG
preconditioner is symmetric if <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img2.png"
ALT="$A$"></SPAN>, <SPAN CLASS="MATH"><IMG
ALT="$A$">, <IMG
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img48.png"
ALT="$M_{1L}$"></SPAN> and <SPAN CLASS="MATH"><IMG
ALT="$M_{1L}$"> and <IMG
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$M_{C}$"></SPAN> are symmetric.
ALT="$M_{C}$"> are symmetric.
<P>
As previously noted, on parallel computers the number of submatrices usually matches
@@ -644,7 +643,7 @@ On the other hand, the use of few processors often leads to local sumatrices tha
are too expensive to be processed on single processors, because of memory and/or
computing requirements. Therefore, it seems natural to use a recursive approach,
in which the coarse-level correction is re-applied starting from the current
coarse-level system. The corresponding preconditioners, called <SPAN CLASS="textit">multi-level</SPAN>
coarse-level system. The corresponding preconditioners, called <I>multi-level</I>
preconditioners, can significantly reduce the computational cost of preconditioning
with respect to the two-level case (see [<A
HREF="node24.html#dd2_96">20</A>, Chapter 3]).
@@ -654,37 +653,37 @@ For a detailed descrition of them, the reader is
referred to [<A
HREF="node24.html#dd2_96">20</A>, Chapter 3].
The algorithm for the application of a multi-level hybrid
post-smoothed preconditioner <SPAN CLASS="MATH"><IMG
post-smoothed preconditioner <IMG
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img59.png"
ALT="$M$"></SPAN> to a vector <SPAN CLASS="MATH"><IMG
ALT="$M$"> to a vector <IMG
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
ALT="$v$"></SPAN>, i.e. for the
computation of <SPAN CLASS="MATH"><IMG
ALT="$v$">, i.e. for the
computation of <IMG
WIDTH="87" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img60.png"
ALT="$w=M^{-1}v$"></SPAN>, is reported, for
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 <SPAN CLASS="MATH"><IMG
is denoted by <IMG
WIDTH="37" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img61.png"
ALT="$nlev$"></SPAN> and the levels are numbered in increasing order starting
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 <SPAN CLASS="MATH"><IMG
and the corresponding basic preconditioner at each level <IMG
WIDTH="10" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img62.png"
ALT="$l$"></SPAN> are denoted by <SPAN CLASS="MATH"><IMG
ALT="$l$"> are denoted by <IMG
WIDTH="22" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img63.png"
ALT="$A_l$"></SPAN> and
<SPAN CLASS="MATH"><IMG
ALT="$A_l$"> and
<IMG
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img64.png"
ALT="$M_l$"></SPAN>, respectively, with <SPAN CLASS="MATH"><IMG
ALT="$M_l$">, respectively, with <IMG
WIDTH="61" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img65.png"
ALT="$A_1=A$"></SPAN>.
ALT="$A_1=A$">.
<DIV ALIGN="CENTER"><A NAME="fig:mlhpost_alg"></A><A NAME="508"></A>
<TABLE>
@@ -728,7 +727,7 @@ $w = y_1$;
\begin{tabbing}
\quad \=\quad...
...= y_l+r_l$\\
\textbf{endfor} [1mm]
\textbf{endfor} \\ [1mm]
$w = y_1$;
\end{tabbing}}
\end{minipage}}">
@@ -738,8 +737,7 @@ $w = y_1$;
</DIV>
<P>
<DIV CLASS="navigation"><HR>
<HR>
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<A NAME="tex2html194"
HREF="node12.html">
@@ -765,11 +763,11 @@ $w = y_1$;
<B> Previous:</B> <A NAME="tex2html185"
HREF="node10.html">Multi-level Domain Decomposition Background</A>
&nbsp; <B> <A NAME="tex2html193"
HREF="node2.html">Contents</A></B> </DIV>
HREF="node2.html">Contents</A></B>
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<ADDRESS>
Salvatore Filippone
2008-07-23
2008-07-24
</ADDRESS>
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