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<!--Converted with LaTeX2HTML 2012 (1.2)
original version by: Nikos Drakos, CBLU, University of Leeds
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<!--Converted with LaTeX2HTML 2017.2 (Released Jan 23, 2017) -->
<HTML>
<HEAD>
<TITLE>Smoothers and coarsest-level solvers</TITLE>
@@ -13,7 +9,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H2><A NAME="SECTION00063000000000000000"></A><A NAME="sec:smoothers"></A>
<BR>
Smoothers and coarsest-level solvers
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
</H2><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The smoothers implemented in MLD2P4 include the Jacobi and block-Jacobi methods,
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The smoothers implemented in MLD2P4 include the Jacobi and block-Jacobi methods,
a hybrid version of the forward and backward Gauss-Seidel methods, and the
additive Schwarz (AS) ones (see, e.g., [<A
HREF="node30.html#Saad_book">20</A>,<A
HREF="node30.html#dd2_96">21</A>]).
</FONT></FONT></FONT>
HREF="node36.html#Saad_book">20</A>,<A
HREF="node36.html#dd2_96">21</A>]).
</BIG></BIG></BIG>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The hybrid Gauss-Seidel
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The hybrid Gauss-Seidel
version is considered because the original Gauss-Seidel method is inherently sequential.
At each iteration of the hybrid version, each parallel process uses the most recent values
of its own local variables and the values of the non-local variables computed at the
previous iteration, obtained by exchanging data with other processes before
the beginning of the current iteration.
</FONT></FONT></FONT>
</BIG></BIG></BIG>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In the AS methods, the index space <IMG
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In the AS methods, the index space <SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$"> is divided into <IMG
ALT="$\Omega^k$"></SPAN> is divided into <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img50.png"
ALT="$m_k$">
subsets <IMG
ALT="$m_k$"></SPAN>
subsets <SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\Omega^k_i$"> of size <IMG
ALT="$\Omega^k_i$"></SPAN> of size <SPAN CLASS="MATH"><IMG
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img52.png"
ALT="$n_{k,i}$">, possibly
overlapping. For each <IMG
ALT="$n_{k,i}$"></SPAN>, possibly
overlapping. For each <SPAN CLASS="MATH"><IMG
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img30.png"
ALT="$i$"> we consider the restriction
ALT="$i$"></SPAN> we consider the restriction
operator <!-- MATH
$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$
-->
<IMG
<SPAN CLASS="MATH"><IMG
WIDTH="110" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img53.png"
ALT="$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$">
that maps a vector <IMG
ALT="$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$"></SPAN>
that maps a vector <SPAN CLASS="MATH"><IMG
WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img54.png"
ALT="$x^k$"> to the vector <IMG
ALT="$x^k$"></SPAN> to the vector <SPAN CLASS="MATH"><IMG
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img55.png"
ALT="$x_i^k$"> made of the components of <IMG
ALT="$x_i^k$"></SPAN> made of the components of <SPAN CLASS="MATH"><IMG
WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img54.png"
ALT="$x^k$">
with indices in <IMG
ALT="$x^k$"></SPAN>
with indices in <SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\Omega^k_i$">, and the prolongation operator
ALT="$\Omega^k_i$"></SPAN>, and the prolongation operator
<!-- MATH
$P^k_i = (R_i^k)^T$
-->
<IMG
<SPAN CLASS="MATH"><IMG
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img56.png"
ALT="$P^k_i = (R_i^k)^T$">. These operators are then used to build
ALT="$P^k_i = (R_i^k)^T$"></SPAN>. These operators are then used to build
<!-- MATH
$A_i^k=R_i^kA^kP_i^k$
-->
<IMG
<SPAN CLASS="MATH"><IMG
WIDTH="113" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img57.png"
ALT="$A_i^k=R_i^kA^kP_i^k$">, which is the restriction of <IMG
ALT="$A_i^k=R_i^kA^kP_i^k$"></SPAN>, which is the restriction of <SPAN CLASS="MATH"><IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$A^k$"> to the index
space <IMG
ALT="$A^k$"></SPAN> to the index
space <SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\Omega^k_i$">.
The classical AS preconditioner <IMG
ALT="$\Omega^k_i$"></SPAN>.
The classical AS preconditioner <SPAN CLASS="MATH"><IMG
WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img58.png"
ALT="$M^k_{AS}$"> is defined as
</FONT></FONT></FONT>
ALT="$M^k_{AS}$"></SPAN> is defined as
</BIG></BIG></BIG>
<BR><P></P>
<DIV ALIGN="CENTER">
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
\begin{displaymath}
( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k},
@@ -149,98 +146,98 @@ The classical AS preconditioner <IMG
ALT="\begin{displaymath}␍ ( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k},␍\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
where <IMG
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
where <SPAN CLASS="MATH"><IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img60.png"
ALT="$A_i^k$"> is supposed to be nonsingular. We observe that an approximate
inverse of <IMG
ALT="$A_i^k$"></SPAN> is supposed to be nonsingular. We observe that an approximate
inverse of <SPAN CLASS="MATH"><IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img60.png"
ALT="$A_i^k$"> is usually considered instead of <IMG
ALT="$A_i^k$"></SPAN> is usually considered instead of <SPAN CLASS="MATH"><IMG
WIDTH="57" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img61.png"
ALT="$(A_i^k)^{-1}$">.
The setup of <IMG
ALT="$(A_i^k)^{-1}$"></SPAN>.
The setup of <SPAN CLASS="MATH"><IMG
WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img58.png"
ALT="$M^k_{AS}$"> during the multilevel build phase
ALT="$M^k_{AS}$"></SPAN> during the multilevel build phase
involves
</FONT></FONT></FONT>
</BIG></BIG></BIG>
<UL>
<LI>the definition of the index subspaces <IMG
<LI>the definition of the index subspaces <SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img62.png"
ALT="$\Omega_i^k$"> and of the corresponding
operators <IMG
ALT="$\Omega_i^k$"></SPAN> and of the corresponding
operators <SPAN CLASS="MATH"><IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img63.png"
ALT="$R_i^k$"> (and <IMG
ALT="$R_i^k$"></SPAN> (and <SPAN CLASS="MATH"><IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img64.png"
ALT="$P_i^k$">);
ALT="$P_i^k$"></SPAN>);
</LI>
<LI>the computation of the submatrices <IMG
<LI>the computation of the submatrices <SPAN CLASS="MATH"><IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img60.png"
ALT="$A_i^k$">;
ALT="$A_i^k$"></SPAN>;
</LI>
<LI>the computation of their inverses (usually approximated
through some form of incomplete factorization).
</LI>
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</UL><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
The computation of <!-- MATH
$z^k=M^k_{AS}w^k$
-->
<IMG
<SPAN CLASS="MATH"><IMG
WIDTH="102" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img65.png"
ALT="$z^k=M^k_{AS}w^k$">, with <!-- MATH
ALT="$z^k=M^k_{AS}w^k$"></SPAN>, with <!-- MATH
$w^k \in \mathbb{R}^{n_k}$
-->
<IMG
<SPAN CLASS="MATH"><IMG
WIDTH="76" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img66.png"
ALT="$w^k \in \mathbb{R}^{n_k}$">, during the
ALT="$w^k \in \mathbb{R}^{n_k}$"></SPAN>, during the
multilevel application phase, requires
</FONT></FONT></FONT>
</BIG></BIG></BIG>
<UL>
<LI>the restriction of <IMG
<LI>the restriction of <SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img67.png"
ALT="$w^k$"> to the subspaces <!-- MATH
ALT="$w^k$"></SPAN> to the subspaces <!-- MATH
$\mathbb{R}^{n_{k,i}}$
-->
<IMG
<SPAN CLASS="MATH"><IMG
WIDTH="41" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img68.png"
ALT="$\mathbb{R}^{n_{k,i}}$">,
ALT="$\mathbb{R}^{n_{k,i}}$"></SPAN>,
i.e. <!-- MATH
$w_i^k = R_i^{k} w^k$
-->
<IMG
<SPAN CLASS="MATH"><IMG
WIDTH="91" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img69.png"
ALT="$w_i^k = R_i^{k} w^k$">;
ALT="$w_i^k = R_i^{k} w^k$"></SPAN>;
</LI>
<LI>the computation of the vectors <!-- MATH
$z_i^k=(A_i^k)^{-1} w_i^k$
-->
<IMG
<SPAN CLASS="MATH"><IMG
WIDTH="119" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img70.png"
ALT="$z_i^k=(A_i^k)^{-1} w_i^k$">;
ALT="$z_i^k=(A_i^k)^{-1} w_i^k$"></SPAN>;
</LI>
<LI>the prolongation and the sum of the previous vectors,
i.e. <!-- MATH
$z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$
-->
<IMG
<SPAN CLASS="MATH"><IMG
WIDTH="127" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img71.png"
ALT="$z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$">.
ALT="$z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$"></SPAN>.
</LI>
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</UL><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
Variants of the classical AS method, which use modifications of the
restriction and prolongation operators, are also implemented in MLD2P4.
Among them, the Restricted AS (RAS) preconditioner usually
@@ -248,41 +245,42 @@ outperforms the classical AS preconditioner in terms of convergence
rate and of computation and communication time on parallel distributed-memory
computers, and is therefore the most widely used among the AS
preconditioners&nbsp;[<A
HREF="node30.html#CAI_SARKIS">6</A>].
</FONT></FONT></FONT>
HREF="node36.html#CAI_SARKIS">6</A>].
</BIG></BIG></BIG>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Direct solvers based on sparse LU factorizations, implemented in the
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">Direct solvers based on sparse LU factorizations, implemented in the
third-party libraries reported in Section&nbsp;<A HREF="node8.html#sec:third-party">3.2</A>, can be applied
as coarsest-level solvers by MLD2P4. Native inexact solvers based on
incomplete LU factorizations, as well as Jacobi, hybrid (forward) Gauss-Seidel,
and block Jacobi preconditioners are also available. Direct solvers usually
lead to more effective preconditioners in terms of algorithmic scalability;
however, this does not guarantee parallel efficiency.
</FONT></FONT></FONT>
</BIG></BIG></BIG>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT></FONT><HR>
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