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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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<B> Next:</B> <A NAME="tex2html242"
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HREF="node14.html">Smoothers and coarsest-level solvers</A>
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HREF="node2.html">Contents</A></B>
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<!--End of Navigation Panel-->
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<H2><A NAME="SECTION00062000000000000000"></A><A NAME="sec:aggregation"></A>
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<BR>
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Smoothed Aggregation
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</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to define the prolongator <IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$">, used to compute
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the coarse-level matrix <IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img15.png"
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ALT="$A^{k+1}$">, MLD2P4 uses the smoothed aggregation
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algorithm described in [<A
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HREF="node29.html#BREZINA_VANEK">2</A>,<A
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HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>].
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The basic idea of this algorithm is to build a coarse set of indices
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<IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img26.png"
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ALT="$\Omega^{k+1}$"> by suitably grouping the indices of <IMG
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WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\Omega^k$"> into disjoint
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subsets (aggregates), and to define the coarse-to-fine space transfer operator
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<IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$"> by applying a suitable smoother to a simple piecewise constant
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prolongation operator, with the aim of improving the quality of the coarse-space correction.
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</FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Three main steps can be identified in the smoothed aggregation procedure:
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</FONT></FONT></FONT>
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<OL>
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<LI>aggregation of the indices of <IMG
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WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\Omega^k$"> to obtain <IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img26.png"
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ALT="$\Omega^{k+1}$">;
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</LI>
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<LI>construction of the prolongator <IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$">;
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</LI>
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<LI>application of <IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$"> and <IMG
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WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img17.png"
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ALT="$R^k=(P^k)^T$"> to build <IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img15.png"
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ALT="$A^{k+1}$">.
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</LI>
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</OL><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to perform the coarsening step, the smoothed aggregation algorithm
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described in [<A
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HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>] is used. In this algorithm,
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each index <!-- MATH
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$j \in \Omega^{k+1}$
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-->
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<IMG
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WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img27.png"
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ALT="$j \in \Omega^{k+1}$"> corresponds to an aggregate <IMG
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WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img28.png"
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ALT="$\Omega^k_j$"> of <IMG
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WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\Omega^k$">,
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consisting of a suitably chosen index <!-- MATH
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$i \in \Omega^k$
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-->
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<IMG
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WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img29.png"
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ALT="$i \in \Omega^k$"> and indices that are (usually) contained in a
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strongly-coupled neighborood of <IMG
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WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img30.png"
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ALT="$i$">, i.e.,
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</FONT></FONT></FONT>
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<BR>
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<DIV ALIGN="RIGHT">
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<!-- MATH
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\begin{equation}
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\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
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\left\{ r \in \Omega^k: |a_{ir}^k| > \theta \sqrt{|a_{ii}^ka_{rr}^k|} \right \} \cup \left\{ i \right\},
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\end{equation}
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-->
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<TABLE WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:strongly_coup"></A><IMG
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WIDTH="387" HEIGHT="72" BORDER="0"
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SRC="img31.png"
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ALT="\begin{displaymath}
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\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
\left\{ r ...
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...vert a_{ii}^ka_{rr}^k\vert} \right \} \cup \left\{ i \right\},
\end{displaymath}"></TD>
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<TD WIDTH=10 ALIGN="RIGHT">
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(3)</TD></TR>
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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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for a given threshold <!-- MATH
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$\theta \in [0,1]$
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-->
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<IMG
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WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img32.png"
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ALT="$\theta \in [0,1]$"> (see [<A
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HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>] for the details).
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Since this algorithm has a sequential nature, a decoupled
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version of it is applied, where each processor independently executes
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the algorithm on the set of indices assigned to it in the initial data
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distribution. This version is embarrassingly parallel, since it does not require any data
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communication. On the other hand, it may produce some nonuniform aggregates
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and is strongly dependent on the number of processors and on the initial partitioning
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of the matrix <IMG
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$A$">. Nevertheless, this parallel algorithm has been chosen for
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MLD2P4, since it has been shown to produce good results in practice
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[<A
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HREF="node29.html#aaecc_07">5</A>,<A
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HREF="node29.html#apnum_07">7</A>,<A
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HREF="node29.html#TUMINARO_TONG">24</A>].
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</FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The prolongator <IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$"> is built starting from a tentative prolongator
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<!-- MATH
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$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$
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-->
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<IMG
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WIDTH="117" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img33.png"
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ALT="$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$">, defined as
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</FONT></FONT></FONT>
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<BR>
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<DIV ALIGN="RIGHT">
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<!-- MATH
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\begin{equation}
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\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
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\left\{ \begin{array}{ll}
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1 & \quad \mbox{if} \; i \in \Omega^k_j, \\
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0 & \quad \mbox{otherwise},
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\end{array} \right.
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\end{equation}
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-->
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<TABLE WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:tent_prol"></A><IMG
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WIDTH="287" HEIGHT="51" BORDER="0"
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SRC="img34.png"
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ALT="\begin{displaymath}
\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
\left\{...
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...ega^k_j, \
0 & \quad \mbox{otherwise},
\end{array} \right.
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\end{displaymath}"></TD>
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<TD WIDTH=10 ALIGN="RIGHT">
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(4)</TD></TR>
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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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where <IMG
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WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img28.png"
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ALT="$\Omega^k_j$"> is the aggregate of <IMG
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WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\Omega^k$">
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corresponding to the index <!-- MATH
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$j \in \Omega^{k+1}$
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-->
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<IMG
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WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img27.png"
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ALT="$j \in \Omega^{k+1}$">.
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<IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$"> is obtained by applying to <IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img35.png"
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ALT="$\bar{P}^k$"> a smoother
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<!-- MATH
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$S^k \in \mathbb{R}^{n_k \times n_k}$
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-->
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<IMG
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WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img36.png"
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ALT="$S^k \in \mathbb{R}^{n_k \times n_k}$">:
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</FONT></FONT></FONT>
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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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P^k = S^k \bar{P}^k,
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\end{displaymath}
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-->
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<IMG
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WIDTH="90" HEIGHT="30" BORDER="0"
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SRC="img37.png"
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ALT="\begin{displaymath}
P^k = S^k \bar{P}^k,
\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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in order to remove nonsmooth components from the range of the prolongator,
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and hence to improve the convergence properties of the multi-level
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method [<A
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HREF="node29.html#BREZINA_VANEK">2</A>,<A
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HREF="node29.html#Stuben_01">23</A>].
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A simple choice for <IMG
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WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img38.png"
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ALT="$S^k$"> is the damped Jacobi smoother:
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</FONT></FONT></FONT>
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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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S^k = I - \omega^k (D^k)^{-1} A^k_F ,
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\end{displaymath}
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-->
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<IMG
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WIDTH="175" HEIGHT="31" BORDER="0"
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SRC="img39.png"
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ALT="\begin{displaymath}
S^k = I - \omega^k (D^k)^{-1} A^k_F ,
\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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where <IMG
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WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img40.png"
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ALT="$D^k$"> is the diagonal matrix with the same diagonal entries as <IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$A^k$">,
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<!-- MATH
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$A^k_F = (\bar{a}_{ij}^k)$
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-->
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<IMG
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WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img42.png"
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ALT="$A^k_F = (\bar{a}_{ij}^k)$"> is the filtered matrix defined as
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</FONT></FONT></FONT>
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<BR>
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<DIV ALIGN="RIGHT">
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<!-- MATH
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\begin{equation}
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\bar{a}_{ij}^k =
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\left \{ \begin{array}{ll}
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a_{ij}^k & \mbox{if } j \in \mathcal{N}_i^k(\theta), \\
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0 & \mbox{otherwise},
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\end{array} \right.
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\; (j \ne i),
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\qquad
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\bar{a}_{ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),
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\end{equation}
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-->
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<TABLE WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:filtered"></A><IMG
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WIDTH="514" HEIGHT="74" BORDER="0"
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SRC="img43.png"
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ALT="\begin{displaymath}
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\bar{a}_{ij}^k =
\left \{ \begin{array}{ll}
a_{ij}^k & ...
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...ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),
\end{displaymath}"></TD>
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<TD WIDTH=10 ALIGN="RIGHT">
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(5)</TD></TR>
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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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and <IMG
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WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
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SRC="img44.png"
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ALT="$\omega^k$"> is an approximation of <IMG
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WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img45.png"
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ALT="$4/(3\rho^k)$">, where
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<IMG
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WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img46.png"
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|
ALT="$\rho^k$"> is the spectral radius of <!-- MATH
|
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|
|
$(D^k)^{-1}A^k_F$
|
|
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|
-->
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<IMG
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|
WIDTH="83" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img47.png"
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|
ALT="$(D^k)^{-1}A^k_F$"> [<A
|
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|
|
HREF="node29.html#BREZINA_VANEK">2</A>].
|
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|
|
In MLD2P4 this approximation is obtained by using <!-- MATH
|
|
|
|
|
$\| A^k_F \|_\infty$
|
|
|
|
|
-->
|
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|
<IMG
|
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|
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
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|
SRC="img48.png"
|
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|
|
|
ALT="$\Vert A^k_F \Vert _\infty$"> as an estimate
|
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|
of <IMG
|
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|
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
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|
SRC="img46.png"
|
|
|
|
|
ALT="$\rho^k$">. Note that for systems coming from uniformly elliptic
|
|
|
|
|
problems, filtering the matrix <IMG
|
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|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img41.png"
|
|
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|
|
ALT="$A^k$"> has little or no effect, and
|
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|
|
<IMG
|
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|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
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|
SRC="img41.png"
|
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|
|
ALT="$A^k$"> can be used instead of <IMG
|
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|
WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img49.png"
|
|
|
|
|
ALT="$A^k_F$">. The latter choice is the default in MLD2P4.
|
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|
|
</FONT></FONT></FONT>
|
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
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