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<H2><A NAME="SECTION00061000000000000000"></A><A NAME="sec:multilevel"></A>
<BR>
Multi-level Schwarz Preconditioners
</H2>
<P>
The Multilevel preconditioners implemented in MLD2P4 are obtained by combining
AS preconditioners with coarse-space corrections; therefore
we first provide a sketch of the AS preconditioners.
<P>
Given the linear system ,
where <!-- MATH
$A=(a_{ij}) \in \Re^{n \times n}$
-->
<IMG
WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
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="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"
ALT="$A$">, where <!-- MATH
$W=\{1, 2, \ldots, n\}$
-->
<IMG
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="img7.png"
ALT="$E=\{(i,j) : a_{ij} \neq 0\}$"> are the vertex set and the edge set of <IMG
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
them. For any integer <IMG
WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img9.png"
ALT="$\delta > 0$">, a <IMG
WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img10.png"
ALT="$\delta$">-overlap
partition of <IMG
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="16" ALIGN="BOTTOM" BORDER="0"
SRC="img11.png"
ALT="$W$">,
i.e. a set of <IMG
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="img13.png"
ALT="$W_i^0 \subset W$"> such that
<!-- MATH
$\cup_{i=1}^m W_i^0 = W$
-->
<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="16" ALIGN="BOTTOM" BORDER="0"
SRC="img10.png"
ALT="$\delta$">-overlap
partition of <IMG
WIDTH="23" HEIGHT="16" 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="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="img16.png"
ALT="$W_i^{\delta-1}$">.
<P>
Let <IMG
WIDTH="22" 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="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="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="45" ALIGN="MIDDLE" BORDER="0"
SRC="img21.png"
ALT="$v_i^{\delta} \in \Re^{n_i^\delta}$">
containing the components of <IMG
WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img22.png"
ALT="$v$"> corresponding to the vertices in
<IMG
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="img23.png"
ALT="$R_i^{\delta}$"> is a
prolongation operator from <!-- MATH
$\Re^{n_i^\delta}$
-->
<IMG
WIDTH="33" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
SRC="img24.png"
ALT="$\Re^{n_i^\delta}$"> to <IMG
WIDTH="26" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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="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="30" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img27.png"
ALT="$W_i^{\delta}$">.
<P>
The <I>classical one-level AS</I> preconditioner is defined by
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
(A_i^\delta)^{-1} R_i^{\delta},
\end{displaymath}
-->
<IMG
WIDTH="207" HEIGHT="58" BORDER="0"
SRC="img28.png"
ALT="\begin{displaymath}
M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
(A_i^\delta)^{-1} R_i^{\delta},
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img29.png"
ALT="$A_i^\delta$"> is assumed to be nonsingular. Its application
to a vector <IMG
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="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="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
$A_i^\delta w_i = v_i$
-->
<IMG
WIDTH="80" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img32.png"
ALT="$A_i^\delta w_i = v_i$">,
<IMG
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="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="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="img35.png"
ALT="$p$">), MILU(<IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img35.png"
ALT="$p$">) and
ILU(<IMG
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img36.png"
ALT="$p,t$">) [<A
HREF="node27.html#Saad_book">22</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 <I>Restricted AS
(RAS)</I> preconditioner [<A
HREF="node27.html#CAI_SARKIS">5</A>,<A
HREF="node27.html#EFSTATHIOU">15</A>]. It
is obtained by zeroing the components of <IMG
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,
which are substituted by <!-- MATH
$(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$
-->
<IMG
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="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="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="img40.png"
ALT="$W_i^\delta \backslash W_i^0$">:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
(A_i^\delta)^{-1} R_i^{\delta}.
\end{displaymath}
-->
<IMG
WIDTH="217" HEIGHT="58" BORDER="0"
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}.
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Analogously, the AS variant called <I>AS with Harmonic extension (ASH)</I>
is defined by
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
(A_i^\delta)^{-1} \tilde{R}_i^0.
\end{displaymath}
-->
<IMG
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.
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
We note that for <IMG
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.
<P>
As already observed, the convergence rate of the one-level Schwarz
preconditioned iterative solvers deteriorates as the number <IMG
WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img12.png"
ALT="$m$"> of partitions
of <IMG
WIDTH="23" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img11.png"
ALT="$W$"> increases [<A
HREF="node27.html#dd1_94">7</A>,<A
HREF="node27.html#dd2_96">23</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 <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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="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="img45.png"
ALT="$W_C$"> of <I>coarse vertices</I>,
with size <IMG
WIDTH="26" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img46.png"
ALT="$n_C$">, and a suitable restriction operator
<!-- MATH
$R_C \in \Re^{n_C \times n}$
-->
<IMG
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="img44.png"
ALT="$A_C$"> is defined as
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A_C=R_C A R_C^T
\end{displaymath}
-->
<IMG
WIDTH="109" HEIGHT="31" BORDER="0"
SRC="img48.png"
ALT="\begin{displaymath}
A_C=R_C A R_C^T
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
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="img49.png"
ALT="$M_{1L}$"> is obtained as
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
M_{C}^{-1}= R_C^T A_C^{-1} R_C,
\end{displaymath}
-->
<IMG
WIDTH="145" HEIGHT="32" BORDER="0"
SRC="img50.png"
ALT="\begin{displaymath}
M_{C}^{-1}= R_C^T A_C^{-1} R_C,
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img44.png"
ALT="$A_C$"> is assumed to be nonsingular. The application of <IMG
WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$M_{C}^{-1}$">
to a vector <IMG
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="img44.png"
ALT="$A_C$">,
may be carried out also approximately.
<P>
The combination of <IMG
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="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
is obtained:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
\end{displaymath}
-->
<IMG
WIDTH="166" HEIGHT="32" BORDER="0"
SRC="img53.png"
ALT="\begin{displaymath}
M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Applying <IMG
WIDTH="59" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img54.png"
ALT="$M_{2L-A}^{-1}$"> to a vector <IMG
WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img22.png"
ALT="$v$"> within a Krylov solver
corresponds to applying <IMG
WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$M_{C}^{-1}$">
and <IMG
WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img55.png"
ALT="$M_{1L}^{-1}$"> to <IMG
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="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img55.png"
ALT="$M_{1L}^{-1}$"> and then
the coarse-level correction operator <IMG
WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$M_{C}^{-1}$">:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\begin{array}{l}
w = M_{1L}^{-1} v, \\
z = w + M_{C}^{-1} (v-Aw);
\end{array}
\end{displaymath}
-->
<IMG
WIDTH="177" HEIGHT="51" BORDER="0"
SRC="img56.png"
ALT="\begin{displaymath}
\begin{array}{l}
w = M_{1L}^{-1} v, \\
z = w + M_{C}^{-1} (v-Aw);
\end{array}\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
this corresponds to the following <I>two-level hybrid pre-smoothed</I>
Schwarz preconditioner:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
\end{displaymath}
-->
<IMG
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}.
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
On the other hand, by applying the smoother after the coarse-level correction,
i.e. by computing
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\begin{array}{l}
w = M_{C}^{-1} v , \\
z = w + M_{1L}^{-1} (v-Aw) ,
\end{array}
\end{displaymath}
-->
<IMG
WIDTH="177" HEIGHT="51" BORDER="0"
SRC="img58.png"
ALT="\begin{displaymath}
\begin{array}{l}
w = M_{C}^{-1} v , \\
z = w + M_{1L}^{-1} (v-Aw) ,
\end{array}\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
the <I>two-level hybrid post-smoothed</I>
Schwarz preconditioner is obtained:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
\end{displaymath}
-->
<IMG
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}.
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<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 <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img2.png"
ALT="$A$">, <IMG
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img49.png"
ALT="$M_{1L}$"> and <IMG
WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img52.png"
ALT="$M_{C}$"> are symmetric.
<P>
As previously noted, on parallel computers the number of submatrices usually matches
the number of available processors. When the size of the system to be preconditioned
is very large, the use of many processors, i.e. of many small submatrices, often
leads to a large coarse-level system, whose solution may be computationally expensive.
On the other hand, the use of few processors often leads to local sumatrices that
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 <I>multi-level</I>
preconditioners, can significantly reduce the computational cost of preconditioning
with respect to the two-level case (see [<A
HREF="node27.html#dd2_96">23</A>, Chapter 3]).
Additive and hybrid multilevel preconditioners
are obtained as direct extensions of the two-level counterparts.
For a detailed descrition of them, the reader is
referred to [<A
HREF="node27.html#dd2_96">23</A>, Chapter 3].
The algorithm for the application of a multi-level hybrid
post-smoothed preconditioner <IMG
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img60.png"
ALT="$M$"> to a vector <IMG
WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img22.png"
ALT="$v$">, i.e. for the
computation of <IMG
WIDTH="86" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img61.png"
ALT="$w=M^{-1}v$">, is reported, for
example, in Figure <A HREF="#fig:mlhpost_alg">1</A>. Here the number of levels
is denoted by <IMG
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="16" ALIGN="BOTTOM" 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="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="23" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img67.png"
ALT="$R_l$">.
<DIV ALIGN="CENTER"><A NAME="fig:mlhpost_alg"></A><A NAME="514"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
Application of the multi-level hybrid post-smoothed preconditioner.</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<!-- MATH
$\framebox{
\begin{minipage}{.85\textwidth} {\small
\begin{tabbing}
\quad \=\quad \=\quad \=\quad \\[-1mm]$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]
\textbf{endfor} \\[2mm]
! apply the coarsest-level correction\\[1mm]
$y_{nlev} = A_{nlev}^{-1} v_{nlev}$\\[2mm]
\textbf{for $l=nlev -1 , 1, -1$\ do}\\[1mm]
\> ! transfer $y_{l+1}$\ to the next finer level\\
\> $y_l = R_{l+1}^T y_{l+1}$;\\[1mm]
\> ! compute the residual at the current level\\
\> $r_l = v_l-A_l^{-1} y_l$;\\[1mm]
\> ! apply the basic Schwarz preconditioner to the residual\\
\> $r_l = M_l^{-1} r_l$\\[1mm]
\> ! update $y_l$\\
\> $y_l = y_l+r_l$\\
\textbf{endfor} \\[1mm]
$w = y_1$;
\end{tabbing}
}
\end{minipage}
}$
-->
<IMG
WIDTH="430" HEIGHT="435" ALIGN="BOTTOM" BORDER="0"
SRC="img68.png"
ALT="\framebox{
\begin{minipage}{.85\textwidth} {\small
\begin{tabbing}
\quad \=\quad...
...= y_l+r_l$\\
\textbf{endfor} [1mm]
$w = y_1$;
\end{tabbing}}
\end{minipage}}">
</DIV></TD></TR>
</TABLE>
</DIV>
<P>
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