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@@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<TITLE>Multi-level Schwarz Preconditioners</TITLE>
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@@ -18,751 +18,168 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<BR>
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HREF="node10.html">Example and test programs</A>
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HREF="node2.html">Contents</A></B>
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<H2><A NAME="SECTION00061000000000000000"></A><A NAME="sec:multilevel"></A>
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<H1><A NAME="SECTION00060000000000000000"></A><A NAME="sec:background"></A>
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<BR>
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Multi-level Schwarz Preconditioners
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</H2>
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Multi-level Domain Decomposition Background
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</H1>
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<P>
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The Multilevel preconditioners implemented in MLD2P4 are obtained by combining
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AS preconditioners with coarse-space corrections; therefore
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we first provide a sketch of the AS preconditioners.
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<I>Domain Decomposition</I> (DD) preconditioners, coupled with Krylov iterative
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solvers, are widely used in the parallel solution of large and sparse linear systems.
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These preconditioners are based on the divide and conquer technique: the matrix
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to be preconditioned is divided into submatrices, a ``local'' linear system
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involving each submatrix is (approximately) solved, and the local solutions are used
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to build a preconditioner for the whole original matrix. This process
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often corresponds to dividing a physical domain associated to the original matrix
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into subdomains, e.g. in a PDE discretization, to (approximately) solving the
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subproblems corresponding to the subdomains and to building an approximate
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solution of the original problem from the local solutions
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[<A
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HREF="node25.html#Cai_Widlund_92">6</A>,<A
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HREF="node25.html#dd1_94">7</A>,<A
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HREF="node25.html#dd2_96">20</A>].
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<P>
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Given the linear system ,
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where <!-- MATH
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$A=(a_{ij}) \in \Re^{n \times n}$
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-->
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<IMG
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WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
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SRC="img3.png"
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ALT="$A=(a_{ij}) \in \Re^{n \times n}$"> is a
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nonsingular sparse matrix with a symmetric nonzero pattern,
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let <IMG
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WIDTH="93" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img4.png"
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ALT="$G=(W,E)$"> be the adjacency graph of <IMG
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img2.png"
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ALT="$A$">, where <!-- MATH
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$W=\{1, 2, \ldots, n\}$
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-->
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<IMG
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WIDTH="138" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.png"
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ALT="$W=\{1, 2, \ldots, n\}$">
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and <!-- MATH
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$E=\{(i,j) : a_{ij} \neq 0\}$
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-->
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<IMG
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WIDTH="162" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img6.png"
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ALT="$E=\{(i,j) : a_{ij} \neq 0\}$"> are the vertex set and the edge set of <IMG
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img7.png"
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ALT="$G$">,
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respectively. Two vertices are called adjacent if there is an edge connecting
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them. For any integer <IMG
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WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img8.png"
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ALT="$\delta > 0$">, a <IMG
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WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\delta$">-overlap
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partition of <IMG
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WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img10.png"
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ALT="$W$"> can be defined recursively as follows.
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Given a 0-overlap (or non-overlapping) partition of <IMG
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WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img10.png"
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ALT="$W$">,
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i.e. a set of <IMG
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WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img11.png"
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ALT="$m$"> disjoint nonempty sets <!-- MATH
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$W_i^0 \subset W$
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-->
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<IMG
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WIDTH="73" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img12.png"
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ALT="$W_i^0 \subset W$"> such that
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<!-- MATH
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$\cup_{i=1}^m W_i^0 = W$
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-->
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<IMG
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WIDTH="107" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img13.png"
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ALT="$\cup_{i=1}^m W_i^0 = W$">, a <IMG
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WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\delta$">-overlap
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partition of <IMG
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WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img10.png"
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ALT="$W$"> is obtained by considering the sets
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<!-- MATH
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$W_i^\delta \supset W_i^{\delta-1}$
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-->
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<IMG
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WIDTH="97" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
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SRC="img14.png"
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ALT="$W_i^\delta \supset W_i^{\delta-1}$"> obtained by including the vertices that
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are adjacent to any vertex in <!-- MATH
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$W_i^{\delta-1}$
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-->
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<IMG
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WIDTH="48" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
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SRC="img15.png"
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ALT="$W_i^{\delta-1}$">.
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<I>Additive Schwarz</I> preconditioners are DD preconditioners using overlapping
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submatrices, i.e. with some common rows, to couple the local information
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related to the submatrices (see, e.g., [<A
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HREF="node25.html#dd2_96">20</A>]).
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The main motivation for choosing Additive Schwarz preconditioners is their
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intrinsic parallelism. A drawback of these
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preconditioners is that the number of iterations of the preconditioned solvers
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generally grows with the number of submatrices. This may be a serious limitation
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on parallel computers, since the number of submatrices usually matches the number
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of available processors. Optimal convergence rates, i.e. iteration numbers
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independent of the number of submatrices, can be obtained by correcting the
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preconditioner through a suitable approximation of the original linear system
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in a coarse space, which globally couples the information related to the single
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submatrices.
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<P>
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Let <IMG
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WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img16.png"
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ALT="$n_i^\delta$"> be the size of <IMG
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WIDTH="30" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img17.png"
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ALT="$W_i^\delta$"> and <!-- MATH
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$R_i^{\delta} \in
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\Re^{n_i^\delta \times n}$
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-->
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<IMG
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WIDTH="93" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
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SRC="img18.png"
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ALT="$R_i^{\delta} \in
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\Re^{n_i^\delta \times n}$"> the restriction operator that maps
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a vector <IMG
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WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img19.png"
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ALT="$v \in \Re^n$"> onto the vector <!-- MATH
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$v_i^{\delta} \in \Re^{n_i^\delta}$
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-->
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<IMG
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WIDTH="70" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
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SRC="img20.png"
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ALT="$v_i^{\delta} \in \Re^{n_i^\delta}$">
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containing the components of <IMG
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WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$v$"> corresponding to the vertices in
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<IMG
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WIDTH="30" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img17.png"
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ALT="$W_i^\delta$">. The transpose of <IMG
|
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WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img22.png"
|
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ALT="$R_i^{\delta}$"> is a
|
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prolongation operator from <!-- MATH
|
||||
$\Re^{n_i^\delta}$
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-->
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<IMG
|
||||
WIDTH="32" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
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SRC="img23.png"
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ALT="$\Re^{n_i^\delta}$"> to <IMG
|
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WIDTH="26" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img24.png"
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ALT="$\Re^n$">.
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The matrix <!-- MATH
|
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$A_i^\delta=R_i^\delta A (R_i^\delta)^T \in
|
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\Re^{n_i^\delta \times n_i^\delta}$
|
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-->
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<IMG
|
||||
WIDTH="201" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img25.png"
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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
|
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img2.png"
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ALT="$A$"> corresponding to the set <IMG
|
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WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img26.png"
|
||||
ALT="$W_i^{\delta}$">.
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<I>Two-level Schwarz</I> preconditioners are obtained
|
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by combining basic (one-level) Schwarz preconditioners with a coarse-level
|
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correction. In this context, the one-level preconditioner is often
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called `smoother'. Different two-level preconditioners are obtained by varying the
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choice of the smoother and of the coarse-level correction, and the
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way they are combined [<A
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||||
HREF="node25.html#dd2_96">20</A>]. The same reasoning can be applied starting
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from the coarse-level system, i.e. a coarse-space correction can be built
|
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from this system, thus obtaining <I>multi-level</I> preconditioners.
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<P>
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The <I>classical one-level AS</I> preconditioner is defined by
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
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(A_i^\delta)^{-1} R_i^{\delta},
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\end{displaymath}
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-->
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<IMG
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WIDTH="206" HEIGHT="58" BORDER="0"
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||||
SRC="img27.png"
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||||
ALT="\begin{displaymath}
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M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
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(A_i^\delta)^{-1} R_i^{\delta},
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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where <IMG
|
||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img28.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"
|
||||
ALT="$v \in \Re^n$"> within a Krylov solver requires the following
|
||||
three steps:
|
||||
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<OL>
|
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<LI>restriction of <IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.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="$i=1,\ldots,m$">;
|
||||
</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="img31.png"
|
||||
ALT="$A_i^\delta w_i = v_i$">,
|
||||
<IMG
|
||||
WIDTH="97" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img30.png"
|
||||
ALT="$i=1,\ldots,m$">;
|
||||
</LI>
|
||||
<LI>prolongation and sum of the <IMG
|
||||
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img32.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"
|
||||
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"
|
||||
ALT="$p$">), MILU(<IMG
|
||||
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img34.png"
|
||||
ALT="$p$">) and
|
||||
ILU(<IMG
|
||||
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img35.png"
|
||||
ALT="$p,t$">) [<A
|
||||
HREF="node24.html#Saad_book">19</A>, Chapter 10].
|
||||
It is worth noting that optimal preconditioners do not necessarily correspond
|
||||
to minimum execution times. Indeed, to obtain effective multi-level preconditioners
|
||||
a tradeoff between optimality of convergence and the cost of building and applying
|
||||
the coarse-space corrections must be achieved. The choice of the number of levels,
|
||||
i.e. of the coarse-space corrections, also affects the effectiveness of the
|
||||
preconditioners. One more goal is to get convergence rates as less sensitive
|
||||
as possible to variations in the matrix coefficients.
|
||||
|
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<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="node24.html#CAI_SARKIS">5</A>,<A
|
||||
HREF="node24.html#EFSTATHIOU">13</A>]. It
|
||||
is obtained by zeroing the components of <IMG
|
||||
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img32.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="118" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img36.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"
|
||||
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="$R_i^\delta$">
|
||||
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"
|
||||
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="216" HEIGHT="58" BORDER="0"
|
||||
SRC="img40.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="218" HEIGHT="58" BORDER="0"
|
||||
SRC="img41.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="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img42.png"
|
||||
ALT="$\delta=0$"> the three variants of the AS preconditioner are
|
||||
all equal to the block-Jacobi preconditioner.
|
||||
Two main approaches can be used to build coarse-space corrections. The geometric approach
|
||||
applies coarsening strategies based on the knowledge of some physical grid associated
|
||||
to the matrix and requires the user to define grid transfer operators from the fine
|
||||
to the coarse levels and vice versa. This may result difficult for complex geometries;
|
||||
furthermore, suitable one-level preconditioners may be required to get efficient
|
||||
interplay between fine and coarse levels, e.g. when matrices with highly varying coefficients
|
||||
are considered. The algebraic approach builds coarse-space corrections using only matrix
|
||||
information. It performs a fully automatic coarsening and enforces the interplay between
|
||||
the fine and coarse levels by suitably choosing the coarse space and the coarse-to-fine
|
||||
interpolation [<A
|
||||
HREF="node25.html#StubenGMD69_99">22</A>].
|
||||
|
||||
<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"
|
||||
ALT="$m$"> of partitions
|
||||
of <IMG
|
||||
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img10.png"
|
||||
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 <IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.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"
|
||||
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="$W_C$"> of <I>coarse vertices</I>,
|
||||
with size <IMG
|
||||
WIDTH="26" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img45.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"
|
||||
ALT="$R_C \in \Re^{n_C \times n}$">, <IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.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="img47.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="img48.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="144" HEIGHT="32" BORDER="0"
|
||||
SRC="img49.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="img43.png"
|
||||
ALT="$A_C$"> is assumed to be nonsingular. The application of <IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img50.png"
|
||||
ALT="$M_{C}^{-1}$">
|
||||
to a vector <IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.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"
|
||||
ALT="$A_C$">,
|
||||
may be carried out also approximately.
|
||||
MLD2P4 uses a pure algebraic approach for building the sequence of coarse matrices
|
||||
starting from the original matrix. The algebraic approach is based on the <I>smoothed
|
||||
aggregation</I> algorithm [<A
|
||||
HREF="node25.html#BREZINA_VANEK">1</A>,<A
|
||||
HREF="node25.html#VANEK_MANDEL_BREZINA">24</A>]. A decoupled version
|
||||
of this algorithm is implemented, where the smoothed aggregation is applied locally
|
||||
to each submatrix [<A
|
||||
HREF="node25.html#TUMINARO_TONG">23</A>]. In the next two subsections we provide
|
||||
a brief description of the multi-level Schwarz preconditioners and of the smoothed
|
||||
aggregation technique as implemented in MLD2P4. For further details the user
|
||||
is referred to [<A
|
||||
HREF="node25.html#para_04">2</A>,<A
|
||||
HREF="node25.html#aaecc_07">3</A>,<A
|
||||
HREF="node25.html#apnum_07">4</A>,<A
|
||||
HREF="node25.html#dd2_96">20</A>].
|
||||
|
||||
<P>
|
||||
The combination of <IMG
|
||||
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img51.png"
|
||||
ALT="$M_{C}$"> and <IMG
|
||||
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img48.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}
|
||||
-->
|
||||
<BR><HR>
|
||||
<!--Table of Child-Links-->
|
||||
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
||||
|
||||
<IMG
|
||||
WIDTH="166" HEIGHT="32" BORDER="0"
|
||||
SRC="img52.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="img53.png"
|
||||
ALT="$M_{2L-A}^{-1}$"> to a vector <IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$"> within a Krylov solver
|
||||
corresponds to applying <IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img50.png"
|
||||
ALT="$M_{C}^{-1}$">
|
||||
and <IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img54.png"
|
||||
ALT="$M_{1L}^{-1}$"> to <IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.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"
|
||||
ALT="$M_{1L}^{-1}$"> and then
|
||||
the coarse-level correction operator <IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img50.png"
|
||||
ALT="$M_{C}^{-1}$">:
|
||||
<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="img55.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="308" HEIGHT="33" BORDER="0"
|
||||
SRC="img56.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="img57.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="317" 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}.
|
||||
\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="img48.png"
|
||||
ALT="$M_{1L}$"> and <IMG
|
||||
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img51.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="node24.html#dd2_96">20</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="node24.html#dd2_96">20</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="img59.png"
|
||||
ALT="$M$"> to a vector <IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$">, i.e. for the
|
||||
computation of <IMG
|
||||
WIDTH="87" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img60.png"
|
||||
ALT="$w=M^{-1}v$">, is reported, for
|
||||
example, in Figure <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"
|
||||
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="$A_l$"> 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"
|
||||
SRC="img65.png"
|
||||
ALT="$A_1=A$">.
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:mlhpost_alg"></A><A NAME="508"></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="img66.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>
|
||||
<UL>
|
||||
<LI><A NAME="tex2html197"
|
||||
HREF="node12.html">Multi-level Schwarz Preconditioners</A>
|
||||
<LI><A NAME="tex2html198"
|
||||
HREF="node13.html">Smoothed Aggregation</A>
|
||||
</UL>
|
||||
<!--End of Table of Child-Links-->
|
||||
<HR>
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html194"
|
||||
<A NAME="tex2html195"
|
||||
HREF="node12.html">
|
||||
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
||||
SRC="file:/usr/share/latex2html/icons/next.png"></A>
|
||||
<A NAME="tex2html190"
|
||||
HREF="node10.html">
|
||||
<A NAME="tex2html191"
|
||||
HREF="userhtml.html">
|
||||
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
||||
SRC="file:/usr/share/latex2html/icons/up.png"></A>
|
||||
<A NAME="tex2html184"
|
||||
<A NAME="tex2html185"
|
||||
HREF="node10.html">
|
||||
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
||||
SRC="file:/usr/share/latex2html/icons/prev.png"></A>
|
||||
<A NAME="tex2html192"
|
||||
<A NAME="tex2html193"
|
||||
HREF="node2.html">
|
||||
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
||||
SRC="file:/usr/share/latex2html/icons/contents.png"></A>
|
||||
<BR>
|
||||
<B> Next:</B> <A NAME="tex2html195"
|
||||
HREF="node12.html">Smoothed Aggregation</A>
|
||||
<B> Up:</B> <A NAME="tex2html191"
|
||||
HREF="node10.html">Multi-level Domain Decomposition Background</A>
|
||||
<B> Previous:</B> <A NAME="tex2html185"
|
||||
HREF="node10.html">Multi-level Domain Decomposition Background</A>
|
||||
<B> <A NAME="tex2html193"
|
||||
<B> Next:</B> <A NAME="tex2html196"
|
||||
HREF="node12.html">Multi-level Schwarz Preconditioners</A>
|
||||
<B> Up:</B> <A NAME="tex2html192"
|
||||
HREF="userhtml.html">userhtml</A>
|
||||
<B> Previous:</B> <A NAME="tex2html186"
|
||||
HREF="node10.html">Example and test programs</A>
|
||||
<B> <A NAME="tex2html194"
|
||||
HREF="node2.html">Contents</A></B>
|
||||
<!--End of Navigation Panel-->
|
||||
|
||||
|
||||
Reference in New Issue
Block a user