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HREF="node10.html">Multi-level Domain Decomposition Background</A>
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HREF="node11.html">Multi-level Schwarz Preconditioners</A>
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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>
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<P>
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In order to define the restriction operator <SPAN CLASS="MATH"><IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img67.png"
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ALT="$R_C$"></SPAN>, which is used to compute
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the coarse-level matrix <SPAN CLASS="MATH"><IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img43.png"
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ALT="$A_C$"></SPAN>, MLD2P4 uses the <SPAN CLASS="textit">smoothed aggregation</SPAN>
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algorithm described in [<A
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HREF="node30.html#BREZINA_VANEK">1</A>,<A
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HREF="node30.html#VANEK_MANDEL_BREZINA">24</A>].
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The basic idea of this algorithm is to build a coarse set of vertices
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<SPAN CLASS="MATH"><IMG
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WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img44.png"
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ALT="$W_C$"></SPAN> by suitably grouping the vertices of <SPAN CLASS="MATH"><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="$W$"></SPAN> into disjoint subsets
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(aggregates), and to define the coarse-to-fine space transfer operator <SPAN CLASS="MATH"><IMG
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WIDTH="29" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
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SRC="img68.png"
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ALT="$R_C^T$"></SPAN> by
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applying a suitable smoother to a simple piecewise constant
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prolongation operator, to improve the quality of the coarse-space correction.
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<P>
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Three main steps can be identified in the smoothed aggregation procedure:
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<OL>
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<LI>coarsening of the vertex set <SPAN CLASS="MATH"><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="$W$"></SPAN>, to obtain <SPAN CLASS="MATH"><IMG
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WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img44.png"
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ALT="$W_C$"></SPAN>;
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</LI>
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<LI>construction of the prolongator <SPAN CLASS="MATH"><IMG
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WIDTH="29" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
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SRC="img68.png"
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ALT="$R_C^T$"></SPAN>;
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</LI>
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<LI>application of <SPAN CLASS="MATH"><IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img67.png"
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ALT="$R_C$"></SPAN> and <SPAN CLASS="MATH"><IMG
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WIDTH="29" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
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SRC="img68.png"
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ALT="$R_C^T$"></SPAN> to build <SPAN CLASS="MATH"><IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img43.png"
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ALT="$A_C$"></SPAN>.
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</LI>
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</OL>
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<P>
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To perform the coarsening step, we have implemented the aggregation algorithm sketched
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in [<A
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HREF="node30.html#apnum_07">4</A>]. According to [<A
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HREF="node30.html#VANEK_MANDEL_BREZINA">24</A>], a modification of
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this algorithm has been actually considered,
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in which each aggregate <SPAN CLASS="MATH"><IMG
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WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img69.png"
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ALT="$N_r$"></SPAN> is made of vertices of <SPAN CLASS="MATH"><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="$W$"></SPAN> that are <SPAN CLASS="textit">strongly coupled</SPAN>
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to a certain root vertex <SPAN CLASS="MATH"><IMG
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WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img70.png"
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ALT="$r \in W$"></SPAN>, i.e. <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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N_r = \left\{s \in W: |a_{rs}| > \theta \sqrt{|a_{rr}a_{ss}|} \right\}
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\cup \left\{ r \right\} ,
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\end{displaymath}
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-->
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<IMG
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WIDTH="319" HEIGHT="38" BORDER="0"
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SRC="img71.png"
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ALT="\begin{displaymath}N_r = \left\{s \in W: \vert a_{rs}\vert > \theta \sqrt{\vert a_{rr}a_{ss}\vert} \right\}
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\cup \left\{ r \right\} ,
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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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for a given <!-- 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="img72.png"
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ALT="$\theta \in [0,1]$"></SPAN>.
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Since this algorithm has a sequential nature, a <SPAN CLASS="textit">decoupled</SPAN> version of
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it has been chosen, where each processor <SPAN CLASS="MATH"><IMG
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WIDTH="10" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img73.png"
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ALT="$i$"></SPAN> independently applies the algorithm to
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the set of vertices <SPAN CLASS="MATH"><IMG
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WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img74.png"
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ALT="$W_i^0$"></SPAN> assigned to it in the initial data distribution. This
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version is embarrassingly parallel, since it does not require any data communication.
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On the other hand, it may produce non-uniform aggregates near boundary vertices,
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i.e. near vertices adjacent to vertices in other processors, and is strongly
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dependent on the number of processors and on the initial partitioning of the matrix <SPAN CLASS="MATH"><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="$A$"></SPAN>.
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Nevertheless, this algorithm has been chosen for the implementation in MLD2P4,
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since it has been shown to produce good results in practice
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[<A
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HREF="node30.html#aaecc_07">3</A>,<A
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HREF="node30.html#apnum_07">4</A>,<A
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HREF="node30.html#TUMINARO_TONG">23</A>].
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<P>
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The prolongator <SPAN CLASS="MATH"><IMG
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WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
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SRC="img75.png"
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ALT="$P_C=R_C^T$"></SPAN> is built starting from a <SPAN CLASS="textit">tentative prolongator</SPAN>
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<!-- MATH
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$P \in \Re^{n \times n_C}$
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-->
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<SPAN CLASS="MATH"><IMG
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WIDTH="90" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
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SRC="img76.png"
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ALT="$P \in \Re^{n \times n_C}$"></SPAN>, defined as
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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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P=(p_{ij}), \quad p_{ij}=
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\left\{ \begin{array}{ll}
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1 & \quad \mbox{if} \; i \in V^j_C \\
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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="290" HEIGHT="52" BORDER="0"
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SRC="img77.png"
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ALT="\begin{displaymath}
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P=(p_{ij}), \quad p_{ij}=
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\left\{ \begin{array}{ll}
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1 & \qu...
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...\in V^j_C \\
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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">2</SPAN>)</TD></TR>
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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P>
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<SPAN CLASS="MATH"><IMG
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WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img78.png"
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ALT="$P_C$"></SPAN> is obtained by
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applying to <SPAN CLASS="MATH"><IMG
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img79.png"
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ALT="$P$"></SPAN> a smoother <!-- MATH
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$S \in \Re^{n \times n}$
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-->
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<SPAN CLASS="MATH"><IMG
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WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$S \in \Re^{n \times n}$"></SPAN>:
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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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P_C = S P,
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\end{equation}
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-->
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<A NAME="eq:smoothed_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:smoothed_prol"></A><IMG
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WIDTH="73" HEIGHT="30" BORDER="0"
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SRC="img81.png"
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ALT="\begin{displaymath}
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P_C = S P,
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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>
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in order to remove oscillatory 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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Schwarz method [<A
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HREF="node30.html#BREZINA_VANEK">1</A>,<A
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HREF="node30.html#StubenGMD69_99">22</A>].
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A simple choice for <SPAN CLASS="MATH"><IMG
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WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img82.png"
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ALT="$S$"></SPAN> is the damped Jacobi smoother:
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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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S = I - \omega D^{-1} A ,
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\end{equation}
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-->
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<A NAME="eq:jac_smoother"></A>
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<TABLE WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:jac_smoother"></A><IMG
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WIDTH="126" HEIGHT="30" BORDER="0"
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SRC="img83.png"
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ALT="\begin{displaymath}
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S = I - \omega D^{-1} A ,
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\end{displaymath}"></TD>
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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>
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where the value of <SPAN CLASS="MATH"><IMG
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WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img84.png"
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ALT="$\omega$"></SPAN> can be chosen
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using some estimate of the spectral radius of <SPAN CLASS="MATH"><IMG
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WIDTH="50" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
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SRC="img85.png"
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ALT="$D^{-1}A$"></SPAN> [<A
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HREF="node30.html#BREZINA_VANEK">1</A>].
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