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<H2><A NAME="SECTION00061000000000000000"></A><A NAME="sec:multilevel"></A>
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<BR>
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AMG preconditioners
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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 describe the AMG preconditioners available in MLD2P4, we consider a
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linear system
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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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Ax=b,
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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:system"></A><IMG
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WIDTH="58" HEIGHT="30" BORDER="0"
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SRC="img2.png"
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ALT="\begin{displaymath}
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Ax=b,
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\end{displaymath}"></TD>
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<TD WIDTH=10 ALIGN="RIGHT">
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(2)</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 <!-- MATH
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$A=(a_{ij}) \in \mathbb{R}^{n \times n}$
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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="img5.png"
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ALT="$A=(a_{ij}) \in \mathbb{R}^{n \times n}$"> is a nonsingular sparse matrix;
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for ease of presentation we assume <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="$A$"> is real, but the
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results are valid for the complex case as well.
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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">Let us assume as finest index space the set of row (column) indices of <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="$A$">, i.e.,
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<!-- MATH
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$\Omega = \{1, 2, \ldots, n\}$
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-->
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<IMG
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WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img6.png"
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ALT="$\Omega = \{1, 2, \ldots, n\}$">.
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Any algebraic multilevel preconditioners implemented in MLD2P4 generates
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a hierarchy of index spaces and a corresponding hierarchy of matrices,
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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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\Omega^1 \equiv \Omega \supset \Omega^2 \supset \ldots \supset \Omega^{nlev},
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\quad A^1 \equiv A, A^2, \ldots, A^{nlev},
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\end{displaymath}
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-->
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<IMG
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WIDTH="398" HEIGHT="30" BORDER="0"
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SRC="img7.png"
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ALT="\begin{displaymath}\Omega^1 \equiv \Omega \supset \Omega^2 \supset \ldots \supset \Omega^{nlev},
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\quad A^1 \equiv A, A^2, \ldots, A^{nlev}, \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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by using the information contained in <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="$A$">, without assuming any
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knowledge of the geometry of the problem from which <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="$A$"> originates.
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A vector space <!-- MATH
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$\mathbb{R}^{n_{k}}$
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-->
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<IMG
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WIDTH="33" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img8.png"
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ALT="$\mathbb{R}^{n_{k}}$"> is associated with <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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where <IMG
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WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
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SRC="img10.png"
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ALT="$n_k$"> is the size 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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For all <IMG
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WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img11.png"
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ALT="$k < nlev$">, a restriction operator and a prolongation one are built,
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which connect two levels <IMG
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WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img12.png"
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ALT="$k$"> and <IMG
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WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img13.png"
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ALT="$k+1$">:
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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 \in \mathbb{R}^{n_k \times n_{k+1}}, \quad
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R^k \in \mathbb{R}^{n_{k+1}\times n_k};
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\end{displaymath}
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-->
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<IMG
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WIDTH="254" HEIGHT="30" BORDER="0"
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SRC="img14.png"
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ALT="\begin{displaymath}
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P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad
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R^k \in \mathbb{R}^{n_{k+1}\times n_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><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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the 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="$A^{k+1}$"> is computed by using the previous operators according
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to the Galerkin approach, i.e.,
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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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A^{k+1}=R^kA^kP^k.
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\end{displaymath}
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-->
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<IMG
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WIDTH="131" HEIGHT="27" BORDER="0"
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SRC="img16.png"
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ALT="\begin{displaymath}
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A^{k+1}=R^kA^kP^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><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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In the current implementation of MLD2P4 we have <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="$R^k=(P^k)^T$">
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A smoother with iteration matrix <IMG
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WIDTH="32" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img18.png"
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ALT="$M^k$"> is set up at each level <IMG
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WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img11.png"
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ALT="$k < nlev$">, and a solver
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is set up at the coarsest level, so that they are ready for application
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(for example, setting up a solver based on the <IMG
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WIDTH="30" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img19.png"
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ALT="$LU$"> factorization means computing
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and storing the <IMG
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WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img20.png"
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ALT="$L$"> and <IMG
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WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$U$"> factors). The construction of the hierarchy of AMG components
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described so far corresponds to the so-called build phase of the preconditioner.
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</FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<DIV ALIGN="CENTER"><A NAME="fig:application_alg"></A><A NAME="517"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
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Application phase of a V-cycle preconditioner.</CAPTION>
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<TR><TD>
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<DIV ALIGN="CENTER">
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<!-- MATH
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$\framebox{
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\begin{minipage}{.85\textwidth}
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\begin{tabbing}
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\quad \=\quad \=\quad \=\quad \\[-3mm]
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procedure V-cycle$\left(k,A^k,b^k,u^k\right)$\ \\[2mm]
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\>if $\left(k \ne nlev \right)$\ then \\[1mm]
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\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]
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\>\> $b^{k+1} = R^{k+1}\left(b^k - A^k u^k\right)$\ \\[1mm]
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\>\> $u^{k+1} =$\ V-cycle$\left(k+1,A^{k+1},b^{k+1},0\right)$\ \\[1mm]
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\>\> $u^k = u^k + P^{k+1} u^{k+1}$\ \\[1mm]
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\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]
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\>else \\[1mm]
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\>\> $u^k = \left(A^k\right)^{-1} b^k$\\[1mm]
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\>endif \\[1mm]
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\>return $u^k$\ \\[1mm]
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end
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\end{tabbing}
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\end{minipage}
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}$
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-->
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<IMG
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WIDTH="333" HEIGHT="336" ALIGN="BOTTOM" BORDER="0"
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SRC="img22.png"
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ALT="\framebox{
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\begin{minipage}{.85\textwidth}
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\begin{tabbing}
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\quad \=\quad \=\quad...
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...mm]
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\>endif \ [1mm]
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\>return $u^k$ \ [1mm]
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end
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\end{tabbing}
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\end{minipage}
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}">
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</DIV></TD></TR>
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</TABLE>
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</DIV>
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<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">The components produced in the build phase may be combined in several ways
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to obtain different multilevel preconditioners;
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this is done in the application phase, i.e., in the computation of a vector
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of type <IMG
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WIDTH="82" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
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SRC="img23.png"
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ALT="$w=B^{-1}v$">, where <IMG
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WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img24.png"
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ALT="$B$"> denotes the preconditioner, usually within an iteration
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of a Krylov solver [<A
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HREF="node29.html#Saad_book">20</A>]. An example of such a combination, known as
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V-cycle, is given in Figure <A HREF="#fig:application_alg">1</A>. In this case, a single iteration
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of the same smoother is used before and after the the recursive call to the V-cycle (i.e.,
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in the pre-smoothing and post-smoothing phases); however, different choices can be
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performed. Other cycles can be defined; in MLD2P4, we implemented the standard V-cycle
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and W-cycle [<A
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HREF="node29.html#Briggs2000">3</A>], and a version of the K-cycle described
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in [<A
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HREF="node29.html#Notay2008">19</A>].
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</FONT></FONT></FONT>
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<P>
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