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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><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
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<P>
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to define the prolongator <SPAN CLASS="MATH"><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="$P^k$"></SPAN>, used to compute
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the coarse-level matrix <SPAN CLASS="MATH"><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="$A^{k+1}$"></SPAN>, MLD2P4 uses the smoothed aggregation
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algorithm described in [<A
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HREF="node36.html#BREZINA_VANEK">2</A>,<A
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HREF="node36.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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<SPAN CLASS="MATH"><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="$\Omega^{k+1}$"></SPAN> by suitably grouping the indices of <SPAN CLASS="MATH"><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="$\Omega^k$"></SPAN> into disjoint
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subsets (aggregates), and to define the coarse-to-fine space transfer operator
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<SPAN CLASS="MATH"><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="$P^k$"></SPAN> 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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</BIG></BIG></BIG>
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<P>
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">Three main steps can be identified in the smoothed aggregation procedure:
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</BIG></BIG></BIG>
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<OL>
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<LI>aggregation of the indices of <SPAN CLASS="MATH"><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="$\Omega^k$"></SPAN> to obtain <SPAN CLASS="MATH"><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="$\Omega^{k+1}$"></SPAN>;
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</LI>
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<LI>construction of the prolongator <SPAN CLASS="MATH"><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="$P^k$"></SPAN>;
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</LI>
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<LI>application of <SPAN CLASS="MATH"><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="$P^k$"></SPAN> and <SPAN CLASS="MATH"><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="$R^k=(P^k)^T$"></SPAN> to build <SPAN CLASS="MATH"><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="$A^{k+1}$"></SPAN>.
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</LI>
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</OL><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
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<P>
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to perform the coarsening step, the smoothed aggregation algorithm
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described in [<A
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HREF="node36.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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<SPAN CLASS="MATH"><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="$j \in \Omega^{k+1}$"></SPAN> corresponds to an aggregate <SPAN CLASS="MATH"><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="$\Omega^k_j$"></SPAN> of <SPAN CLASS="MATH"><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="$\Omega^k$"></SPAN>,
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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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<SPAN CLASS="MATH"><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="$i \in \Omega^k$"></SPAN> and indices that are (usually) contained in a
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strongly-coupled neighborood of <SPAN CLASS="MATH"><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="$i$"></SPAN>, i.e.,
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</BIG></BIG></BIG>
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<BR>
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<DIV ALIGN="RIGHT" CLASS="mathdisplay">
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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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<A NAME="eq:strongly_coup"></A>
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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) =
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\left\{ r ...
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...vert a_{ii}^ka_{rr}^k\vert} \right \} \cup \left\{ i \right\},
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\end{displaymath}"></TD>
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<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
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(<SPAN CLASS="arabic">3</SPAN>)</TD></TR>
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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
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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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<SPAN CLASS="MATH"><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="$\theta \in [0,1]$"></SPAN> (see [<A
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HREF="node36.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 <SPAN CLASS="MATH"><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="$A$"></SPAN>. 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="node36.html#aaecc_07">5</A>,<A
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HREF="node36.html#apnum_07">7</A>,<A
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HREF="node36.html#TUMINARO_TONG">24</A>].
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</BIG></BIG></BIG>
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<P>
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The prolongator <SPAN CLASS="MATH"><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="$P^k$"></SPAN> 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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<SPAN CLASS="MATH"><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="$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$"></SPAN>, defined as
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</BIG></BIG></BIG>
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<BR>
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<DIV ALIGN="RIGHT" CLASS="mathdisplay">
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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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<A NAME="eq:tent_prol"></A>
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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}
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\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
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\left\{...
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...ega^k_j,
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0 & \quad \mbox{otherwise},
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\end{array} \right.
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\end{displaymath}"></TD>
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<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
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(<SPAN CLASS="arabic">4</SPAN>)</TD></TR>
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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
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where <SPAN CLASS="MATH"><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="$\Omega^k_j$"></SPAN> is the aggregate of <SPAN CLASS="MATH"><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="$\Omega^k$"></SPAN>
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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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<SPAN CLASS="MATH"><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="$j \in \Omega^{k+1}$"></SPAN>.
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<SPAN CLASS="MATH"><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="$P^k$"></SPAN> is obtained by applying to <SPAN CLASS="MATH"><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="$\bar{P}^k$"></SPAN> 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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<SPAN CLASS="MATH"><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="$S^k \in \mathbb{R}^{n_k \times n_k}$"></SPAN>:
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</BIG></BIG></BIG>
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<BR><P></P>
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<DIV ALIGN="CENTER" CLASS="mathdisplay">
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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}
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P^k = S^k \bar{P}^k,
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
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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 multilevel
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method [<A
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HREF="node36.html#BREZINA_VANEK">2</A>,<A
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HREF="node36.html#Stuben_01">23</A>].
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A simple choice for <SPAN CLASS="MATH"><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="$S^k$"></SPAN> is the damped Jacobi smoother:
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</BIG></BIG></BIG>
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<BR><P></P>
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<DIV ALIGN="CENTER" CLASS="mathdisplay">
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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}
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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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</DIV>
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<BR CLEAR="ALL">
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<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
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where <SPAN CLASS="MATH"><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="$D^k$"></SPAN> is the diagonal matrix with the same diagonal entries as <SPAN CLASS="MATH"><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="$A^k$"></SPAN>,
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<!-- MATH
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$A^k_F = (\bar{a}_{ij}^k)$
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-->
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<SPAN CLASS="MATH"><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="$A^k_F = (\bar{a}_{ij}^k)$"></SPAN> is the filtered matrix defined as
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</BIG></BIG></BIG>
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<BR>
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<DIV ALIGN="RIGHT" CLASS="mathdisplay">
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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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<A NAME="eq:filtered"></A>
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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 =
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\left \{ \begin{array}{ll}
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a_{ij}^k & ...
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...ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),
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\end{displaymath}"></TD>
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<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
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(<SPAN CLASS="arabic">5</SPAN>)</TD></TR>
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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
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and <SPAN CLASS="MATH"><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="$\omega^k$"></SPAN> is an approximation of <SPAN CLASS="MATH"><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="$4/(3\rho^k)$"></SPAN>, where
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<SPAN CLASS="MATH"><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="$\rho^k$"></SPAN> is the spectral radius of <!-- MATH
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$(D^k)^{-1}A^k_F$
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-->
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<SPAN CLASS="MATH"><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="$(D^k)^{-1}A^k_F$"></SPAN> [<A
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HREF="node36.html#BREZINA_VANEK">2</A>].
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In MLD2P4 this approximation is obtained by using <!-- MATH
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$\| A^k_F \|_\infty$
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-->
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<SPAN CLASS="MATH"><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="$\Vert A^k_F \Vert _\infty$"></SPAN> as an estimate
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of <SPAN CLASS="MATH"><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="$\rho^k$"></SPAN>. Note that for systems coming from uniformly elliptic
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problems, filtering the matrix <SPAN CLASS="MATH"><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="$A^k$"></SPAN> has little or no effect, and
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<SPAN CLASS="MATH"><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="$A^k$"></SPAN> can be used instead of <SPAN CLASS="MATH"><IMG
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WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img49.png"
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ALT="$A^k_F$"></SPAN>. The latter choice is the default in MLD2P4.
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