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<H2><A NAME="SECTION00063000000000000000"></A><A NAME="sec:smoothers"></A>
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<BR>
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Smoothers and coarsest-level solvers
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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">The smoothers implemented in MLD2P4 include the Jacobi and block-Jacobi methods,
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a hybrid version of the forward and backward Gauss-Seidel methods, and the
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additive Schwarz (AS) ones (see, e.g., [<A
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HREF="node30.html#Saad_book">20</A>,<A
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HREF="node30.html#dd2_96">21</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 hybrid Gauss-Seidel
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version is considered because the original Gauss-Seidel method is inherently sequential.
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At each iteration of the hybrid version, each parallel process uses the most recent values
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of its own local variables and the values of the non-local variables computed at the
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previous iteration, obtained by exchanging data with other processes before
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the beginning of the current iteration.
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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">In the AS methods, the index space <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="$\Omega^k$"> is divided into <IMG
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WIDTH="28" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
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SRC="img50.png"
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ALT="$m_k$">
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subsets <IMG
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WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img51.png"
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ALT="$\Omega^k_i$"> of size <IMG
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WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
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SRC="img52.png"
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ALT="$n_{k,i}$">, possibly
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overlapping. For each <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="$i$"> we consider the restriction
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operator <!-- MATH
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$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$
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-->
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<IMG
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WIDTH="110" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img53.png"
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ALT="$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$">
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that maps a vector <IMG
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WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img54.png"
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ALT="$x^k$"> to the vector <IMG
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WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img55.png"
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ALT="$x_i^k$"> made of the components of <IMG
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WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img54.png"
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ALT="$x^k$">
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with indices in <IMG
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WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img51.png"
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ALT="$\Omega^k_i$">, and the prolongation operator
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<!-- MATH
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$P^k_i = (R_i^k)^T$
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-->
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<IMG
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WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img56.png"
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ALT="$P^k_i = (R_i^k)^T$">. These operators are then used to build
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<!-- MATH
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$A_i^k=R_i^kA^kP_i^k$
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-->
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<IMG
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WIDTH="113" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img57.png"
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ALT="$A_i^k=R_i^kA^kP_i^k$">, which is the restriction of <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="$A^k$"> to the index
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space <IMG
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WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img51.png"
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ALT="$\Omega^k_i$">.
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The classical AS preconditioner <IMG
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WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img58.png"
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ALT="$M^k_{AS}$"> is defined as
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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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( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k},
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\end{displaymath}
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-->
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<IMG
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WIDTH="219" HEIGHT="59" BORDER="0"
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SRC="img59.png"
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ALT="\begin{displaymath}
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( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{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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where <IMG
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WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img60.png"
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ALT="$A_i^k$"> is supposed to be nonsingular. We observe that an approximate
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inverse of <IMG
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WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img60.png"
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ALT="$A_i^k$"> is usually considered instead of <IMG
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WIDTH="57" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img61.png"
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ALT="$(A_i^k)^{-1}$">.
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The setup of <IMG
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WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img58.png"
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ALT="$M^k_{AS}$"> during the multilevel build phase
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involves
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</FONT></FONT></FONT>
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<UL>
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<LI>the definition of the index subspaces <IMG
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WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img62.png"
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ALT="$\Omega_i^k$"> and of the corresponding
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operators <IMG
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WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img63.png"
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ALT="$R_i^k$"> (and <IMG
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WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img64.png"
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ALT="$P_i^k$">);
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</LI>
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<LI>the computation of the submatrices <IMG
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WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img60.png"
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ALT="$A_i^k$">;
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</LI>
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<LI>the computation of their inverses (usually approximated
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through some form of incomplete factorization).
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</LI>
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</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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The computation of <!-- MATH
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$z^k=M^k_{AS}w^k$
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-->
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<IMG
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WIDTH="102" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img65.png"
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ALT="$z^k=M^k_{AS}w^k$">, with <!-- MATH
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$w^k \in \mathbb{R}^{n_k}$
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-->
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<IMG
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WIDTH="76" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img66.png"
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ALT="$w^k \in \mathbb{R}^{n_k}$">, during the
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multilevel application phase, requires
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</FONT></FONT></FONT>
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<UL>
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<LI>the restriction of <IMG
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WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img67.png"
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ALT="$w^k$"> to the subspaces <!-- MATH
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$\mathbb{R}^{n_{k,i}}$
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-->
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<IMG
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WIDTH="41" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img68.png"
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ALT="$\mathbb{R}^{n_{k,i}}$">,
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i.e. <!-- MATH
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$w_i^k = R_i^{k} w^k$
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-->
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<IMG
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WIDTH="91" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img69.png"
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ALT="$w_i^k = R_i^{k} w^k$">;
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</LI>
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<LI>the computation of the vectors <!-- MATH
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$z_i^k=(A_i^k)^{-1} w_i^k$
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-->
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<IMG
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WIDTH="119" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img70.png"
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ALT="$z_i^k=(A_i^k)^{-1} w_i^k$">;
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</LI>
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<LI>the prolongation and the sum of the previous vectors,
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i.e. <!-- MATH
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$z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$
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-->
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<IMG
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WIDTH="127" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img71.png"
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ALT="$z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$">.
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</LI>
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</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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Variants of the classical AS method, which use modifications of the
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restriction and prolongation operators, are also implemented in MLD2P4.
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Among them, the Restricted AS (RAS) preconditioner usually
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outperforms the classical AS preconditioner in terms of convergence
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rate and of computation and communication time on parallel distributed-memory
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computers, and is therefore the most widely used among the AS
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preconditioners [<A
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HREF="node30.html#CAI_SARKIS">6</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">Direct solvers based on sparse LU factorizations, implemented in the
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third-party libraries reported in Section <A HREF="node8.html#sec:third-party">3.2</A>, can be applied
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as coarsest-level solvers by MLD2P4. Native inexact solvers based on
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incomplete LU factorizations, as well as Jacobi, hybrid (forward) Gauss-Seidel,
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and block Jacobi preconditioners are also available. Direct solvers usually
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lead to more effective preconditioners in terms of algorithmic scalability;
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however, this does not guarantee parallel efficiency.
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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"></FONT></FONT></FONT><HR>
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