</H1><FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">
Multigrid preconditioners, coupled with Krylov iterative
solvers, are widely used in the parallel solution of large and sparse linear systems,
because of their optimality in the solution of linear systems arising from the
discretization of scalar elliptic Partial Differential Equations (PDEs) on regular grids.
Optimality, also known as algorithmic scalability, is the property
of having a computational cost per iteration that depends linearly on
the problem size, and a convergence rate that is independent of the problem size.
Multigrid preconditioners are based on a recursive application of a two-grid process
consisting of smoother iterations and a coarse-space (or coarse-level) correction.
The smoothers may be either basic iterative methods, such as the Jacobi and Gauss-Seidel ones,
or more complex subspace-correction methods, such as the Schwarz ones.
The coarse-space correction consists of solving, in an appropriately chosen
coarse space, the residual equation associated with the approximate solution computed
by the smoother, and of using the solution of this equation to correct the
previous approximation. The transfer of information between the original
(fine) space and the coarse one is performed by using suitable restriction and
prolongation operators. The construction of the coarse space and the corresponding
transfer operators is carried out by applying a so-called coarsening algorithm to the system
matrix. Two main approaches can be used to perform coarsening: the geometric approach,
which exploits the knowledge of some physical grid associated with the matrix
and requires the user to define transfer operators from the fine
to the coarse level and vice versa, and the algebraic approach, which builds
the coarse-space correction and the associate transfer operators using only matrix
information. The first approach may be difficult when the system comes from
discretizations on complex geometries;
furthermore, ad hoc one-level smoothers may be required to get an efficient
interplay between fine and coarse levels, e.g., when matrices with highly varying coefficients
are considered. The second approach performs a fully automatic coarsening and enforces the
interplay between fine and coarse level by suitably choosing the coarse space and
the coarse-to-fine interpolation (see, e.g., [<A
<FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">Multigrid preconditioners, coupled with Krylov iterative
solvers, are widely used in the parallel solution of large and sparse linear systems,
because of their optimality in the solution of linear systems arising from the
discretization of scalar elliptic Partial Differential Equations (PDEs) on regular grids.
Optimality, also known as algorithmic scalability, is the property
of having a computational cost per iteration that depends linearly on
the problem size, and a convergence rate that is independent of the problem size.
</FONT></FONT></FONT>
<P>
<FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">Multigrid preconditioners are based on a recursive application of a two-grid process
consisting of smoother iterations and a coarse-space (or coarse-level) correction.
The smoothers may be either basic iterative methods, such as the Jacobi and Gauss-Seidel ones,
or more complex subspace-correction methods, such as the Schwarz ones.
The coarse-space correction consists of solving, in an appropriately chosen
coarse space, the residual equation associated with the approximate solution computed
by the smoother, and of using the solution of this equation to correct the
previous approximation. The transfer of information between the original
(fine) space and the coarse one is performed by using suitable restriction and
prolongation operators. The construction of the coarse space and the corresponding
transfer operators is carried out by applying a so-called coarsening algorithm to the system
matrix. Two main approaches can be used to perform coarsening: the geometric approach,
which exploits the knowledge of some physical grid associated with the matrix
and requires the user to define transfer operators from the fine
to the coarse level and vice versa, and the algebraic approach, which builds
the coarse-space correction and the associate transfer operators using only matrix
information. The first approach may be difficult when the system comes from
discretizations on complex geometries;
furthermore, ad hoc one-level smoothers may be required to get an efficient
interplay between fine and coarse levels, e.g., when matrices with highly varying coefficients
are considered. The second approach performs a fully automatic coarsening and enforces the
interplay between fine and coarse level by suitably choosing the coarse space and
the coarse-to-fine interpolation (see, e.g., [<A
HREF="node29.html#Briggs2000">3</A>,<A
HREF="node29.html#Stuben_01">23</A>,<A
HREF="node29.html#dd2_96">21</A>] for details.)
MLD2P4 uses a pure algebraic approach, based on the smoothed
aggregation algorithm [<A
HREF="node29.html#dd2_96">21</A>] for details.)
</FONT></FONT></FONT>
<P>
<FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">MLD2P4 uses a pure algebraic approach, based on the smoothed
aggregation algorithm [<A
HREF="node29.html#BREZINA_VANEK">2</A>,<A
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>],
for building the sequence of coarse matrices and transfer operators,
starting from the original one.
A decoupled version of this algorithm is implemented, where the smoothed
aggregation is applied locally to each submatrix [<A
HREF="node29.html#TUMINARO_TONG">24</A>].
A brief description of the AMG preconditioners implemented in MLD2P4 is given in
Sections <AHREF="node12.html#sec:multilevel">4.1</A>-<AHREF="node14.html#sec:smoothers">4.3</A>. For further details the reader
is referred to [<A
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>],
for building the sequence of coarse matrices and transfer operators,
starting from the original one.
A decoupled version of this algorithm is implemented, where the smoothed
aggregation is applied locally to each submatrix [<A
HREF="node29.html#TUMINARO_TONG">24</A>].
A brief description of the AMG preconditioners implemented in MLD2P4 is given in
Sections <AHREF="node12.html#sec:multilevel">4.1</A>-<AHREF="node14.html#sec:smoothers">4.3</A>. For further details the reader
is referred to [<A
HREF="node29.html#para_04">4</A>,<A
HREF="node29.html#aaecc_07">5</A>,<A
HREF="node29.html#apnum_07">7</A>,<A
HREF="node29.html#MLD2P4_TOMS">8</A>].
We note that optimal multigrid preconditioners do not necessarily correspond
to minimum execution times in a parallel setting. Indeed, to obtain effective parallel
multigrid preconditioners, a tradeoff between the optimality and the cost of building and
applying the smoothers and the coarse-space corrections must be achieved. Effective
parallel preconditioners require algorithmic scalability to be coupled with implementation
scalability, i.e., a computational cost per iteration which remains (almost) constant as
the number of parallel processors increases.
</FONT></FONT></FONT>
HREF="node29.html#MLD2P4_TOMS">8</A>].
</FONT></FONT></FONT>
<P>
<FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">We note that optimal multigrid preconditioners do not necessarily correspond
to minimum execution times in a parallel setting. Indeed, to obtain effective parallel
multigrid preconditioners, a tradeoff between the optimality and the cost of building and
applying the smoothers and the coarse-space corrections must be achieved. Effective
parallel preconditioners require algorithmic scalability to be coupled with implementation
scalability, i.e., a computational cost per iteration which remains (almost) constant as
</H2><FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">
In order to describe the AMG preconditioners available in MLD2P4, we consider a
linear system
</FONT></FONT></FONT>
</H2><FONTSIZE="+1"><FONTSIZE="+1"></FONT></FONT>
<P>
<FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">In order to describe the AMG preconditioners available in MLD2P4, we consider a
linear system
</FONT></FONT></FONT>
<BR>
<DIVALIGN="RIGHT">
@ -73,30 +77,41 @@ Ax=b,
<TDWIDTH=10ALIGN="RIGHT">
(2)</TD></TR>
</TABLE>
<BRCLEAR="ALL"></DIV><P></P><FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">
where <!-- MATH
ALT="$A=(a_{ij}) \in \mathbb{R}^{n \times n}$"> is a nonsingular sparse matrix;
for ease of presentation we assume <IMG
ALT="$A=(a_{ij}) \in \mathbb{R}^{n \times n}$"> is a nonsingular sparse matrix;
for ease of presentation we assume <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$"> is real, but the
results are valid for the complex case as well.
Let us assume as finest index space the set of row (column) indices of <IMG
ALT="$A$"> is real, but the
results are valid for the complex case as well.
</FONT></FONT></FONT>
<P>
<FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">Let us assume as finest index space the set of row (column) indices of <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$">, i.e.,
<!-- MATH
ALT="$A$">, i.e.,
<!-- MATH
$\Omega = \{1, 2, \ldots, n\}$
-->
<IMG
WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img6.png"
ALT="$\Omega = \{1, 2, \ldots, n\}$">.
Any algebraic multilevel preconditioners implemented in MLD2P4 generates
a hierarchy of index spaces and a corresponding hierarchy of matrices,
</FONT></FONT></FONT>
ALT="$\Omega = \{1, 2, \ldots, n\}$">.
Any algebraic multilevel preconditioners implemented in MLD2P4 generates
a hierarchy of index spaces and a corresponding hierarchy of matrices,
In the current implementation of MLD2P4 we have <IMG
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img17.png"
ALT="$R^k=(P^k)^T$">
A smoother with iteration matrix <IMG
ALT="$R^k=(P^k)^T$">
A smoother with iteration matrix <IMG
WIDTH="32" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img18.png"
ALT="$M^k$"> is set up at each level <IMG
WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img11.png"
ALT="$k < nlev$">, and a solver
is set up at the coarsest level, so that they are ready for application
(for example, setting up a solver based on the <IMG
ALT="$k < nlev$">, and a solver
is set up at the coarsest level, so that they are ready for application
(for example, setting up a solver based on the <IMG
WIDTH="30" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img19.png"
ALT="$LU$"> factorization means computing
and storing the <IMG
ALT="$LU$"> factorization means computing
and storing the <IMG
WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img20.png"
ALT="$L$"> and <IMG
WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
ALT="$U$"> factors). The construction of the hierarchy of AMG components
described so far corresponds to the so-called build phase of the preconditioner.
</FONT></FONT></FONT>
ALT="$U$"> factors). The construction of the hierarchy of AMG components
described so far corresponds to the so-called build phase of the preconditioner.
<FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">
The components produced in the build phase may be combined in several ways
to obtain different multilevel preconditioners;
this is done in the application phase, i.e., in the computation of a vector
of type <IMG
<FONTSIZE="+1"><FONTSIZE="+1"></FONT></FONT>
<P>
<FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">The components produced in the build phase may be combined in several ways
to obtain different multilevel preconditioners;
this is done in the application phase, i.e., in the computation of a vector
of type <IMG
WIDTH="82" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img23.png"
ALT="$w=B^{-1}v$">, where <IMG
WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img24.png"
ALT="$B$"> denotes the preconditioner, usually within an iteration
of a Krylov solver [<A
HREF="node29.html#Saad_book">20</A>]. An example of such a combination, known as
V-cycle, is given in Figure <AHREF="#fig:application_alg">1</A>. In this case, a single iteration
of the same smoother is used before and after the the recursive call to the V-cycle (i.e.,
in the pre-smoothing and post-smoothing phases); however, different choices can be
performed. Other cycles can be defined; in MLD2P4, we implemented the standard V-cycle
and W-cycle [<A
HREF="node29.html#Briggs2000">3</A>], and a version of the K-cycle described
in [<A
</H2><FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">
In order to define the prolongator <IMG
</H2><FONTSIZE="+1"><FONTSIZE="+1"></FONT></FONT>
<P>
<FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">In order to define the prolongator <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$">, used to compute
the coarse-level matrix <IMG
ALT="$P^k$">, used to compute
the coarse-level matrix <IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img15.png"
ALT="$A^{k+1}$">, MLD2P4 uses the smoothed aggregation
algorithm described in [<A
ALT="$A^{k+1}$">, MLD2P4 uses the smoothed aggregation
algorithm described in [<A
HREF="node29.html#BREZINA_VANEK">2</A>,<A
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>].
The basic idea of this algorithm is to build a coarse set of indices
<IMG
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>].
The basic idea of this algorithm is to build a coarse set of indices
<IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img26.png"
ALT="$\Omega^{k+1}$"> by suitably grouping the indices of <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$"> into disjoint
subsets (aggregates), and to define the coarse-to-fine space transfer operator
<IMG
ALT="$\Omega^k$"> into disjoint
subsets (aggregates), and to define the coarse-to-fine space transfer operator
<IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"> by applying a suitable smoother to a simple piecewise constant
prolongation operator, with the aim of improving the quality of the coarse-space correction.
Three main steps can be identified in the smoothed aggregation procedure:
</FONT></FONT></FONT>
ALT="$P^k$"> by applying a suitable smoother to a simple piecewise constant
prolongation operator, with the aim of improving the quality of the coarse-space correction.
</FONT></FONT></FONT>
<P>
<FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">Three main steps can be identified in the smoothed aggregation procedure:
</FONT></FONT></FONT>
<OL>
<LI>aggregation of the indices of <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
@ -80,12 +92,12 @@ Smoothed Aggregation
ALT="$\Omega^k$"> to obtain <IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img26.png"
ALT="$\Omega^{k+1}$">;
ALT="$\Omega^{k+1}$">;
</LI>
<LI>construction of the prolongator <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$">;
ALT="$P^k$">;
</LI>
<LI>application of <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
@ -96,10 +108,14 @@ Smoothed Aggregation
ALT="$R^k=(P^k)^T$"> to build <IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img15.png"
ALT="$A^{k+1}$">.
ALT="$A^{k+1}$">.
</LI>
</OL><FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">
In order to perform the coarsening step, the smoothed aggregation algorithm
described in [<A
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>] is used. In this algorithm,
each index <!-- MATH
</OL><FONTSIZE="+1"><FONTSIZE="+1"></FONT></FONT>
<P>
<FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">In order to perform the coarsening step, the smoothed aggregation algorithm
described in [<A
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>] is used. In this algorithm,
each index <!-- MATH
$j \in \Omega^{k+1}$
-->
<IMG
@ -111,22 +127,26 @@ Smoothed Aggregation
ALT="$\Omega^k_j$"> of <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$">,
consisting of a suitably chosen index <!-- MATH
ALT="$\Omega^k$">,
consisting of a suitably chosen index <!-- MATH
$i \in \Omega^k$
-->
<IMG
WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img29.png"
ALT="$i \in \Omega^k$"> and indices that are (usually) contained in a
strongly-coupled neighborood of <IMG
ALT="$i \in \Omega^k$"> and indices that are (usually) contained in a
strongly-coupled neighborood of <IMG
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img30.png"
ALT="$i$">, i.e.,
</FONT></FONT></FONT>
ALT="$i$">, i.e.,
</FONT></FONT></FONT>
<BR>
<DIVALIGN="RIGHT">
<!-- MATH
\begin{equation}
\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
\left\{ r \in \Omega^k: |a_{ir}^k| > \theta \sqrt{|a_{ii}^ka_{rr}^k|} \right \} \cup \left\{ i \right\},
\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
\left\{ r \in \Omega^k: |a_{ir}^k| > \theta \sqrt{|a_{ii}^ka_{rr}^k|} \right \} \cup \left\{ i \right\},
\end{equation}
-->
<TABLEWIDTH="100%"ALIGN="CENTER">
@ -139,35 +159,54 @@ Smoothed Aggregation
<TDWIDTH=10ALIGN="RIGHT">
(3)</TD></TR>
</TABLE>
<BRCLEAR="ALL"></DIV><P></P><FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">
for a given threshold <!-- MATH
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>] for the details).
Since this algorithm has a sequential nature, a decoupled
version of it is applied, where each processor independently executes
the algorithm on the set of indices assigned to it in the initial data
distribution. This version is embarrassingly parallel, since it does not require any data
communication. On the other hand, it may produce some nonuniform aggregates
and is strongly dependent on the number of processors and on the initial partitioning
of the matrix <IMG
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>] for the details).
Since this algorithm has a sequential nature, a decoupled
version of it is applied, where each processor independently executes
the algorithm on the set of indices assigned to it in the initial data
distribution. This version is embarrassingly parallel, since it does not require any data
communication. On the other hand, it may produce some nonuniform aggregates
and is strongly dependent on the number of processors and on the initial partitioning
of the matrix <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$">. Nevertheless, this parallel algorithm has been chosen for
MLD2P4, since it has been shown to produce good results in practice
[<A
ALT="$A$">. Nevertheless, this parallel algorithm has been chosen for
MLD2P4, since it has been shown to produce good results in practice
[<A
HREF="node29.html#aaecc_07">5</A>,<A
HREF="node29.html#apnum_07">7</A>,<A
HREF="node29.html#TUMINARO_TONG">24</A>].
The prolongator <IMG
<P></P><FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">
in order to remove nonsmooth components from the range of the prolongator,
and hence to improve the convergence properties of the multi-level
method [<A
</H2><FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">
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
</H2><FONTSIZE="+1"><FONTSIZE="+1"></FONT></FONT>
<P>
<FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">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="node29.html#Saad_book">20</A>,<A
HREF="node29.html#dd2_96">21</A>]).
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.
In the AS methods, the index space <IMG
@ -169,8 +201,9 @@ Smoothers and coarsest-level solvers
<IMG
WIDTH="76" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img66.png"
ALT="$w^k \in \mathbb{R}^{n_k}$">, during the
multilevel application phase, requires
</FONT></FONT></FONT>
ALT="$w^k \in \mathbb{R}^{n_k}$">, during the
multilevel application phase, requires
</FONT></FONT></FONT>
<UL>
<LI>the restriction of <IMG
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
@ -181,13 +214,14 @@ Smoothers and coarsest-level solvers
<IMG
WIDTH="41" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img68.png"
ALT="$\mathbb{R}^{n_{k,i}}$">,
i.e. <!-- MATH
ALT="$\mathbb{R}^{n_{k,i}}$">,
i.e. <!-- MATH
$w_i^k = R_i^{k} w^k$
-->
<IMG
WIDTH="91" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img69.png"
ALT="$w_i^k = R_i^{k} w^k$">;
ALT="$w_i^k = R_i^{k} w^k$">;
</LI>
<LI>the computation of the vectors <!-- MATH
$z_i^k=(A_i^k)^{-1} w_i^k$
@ -195,19 +229,38 @@ Smoothers and coarsest-level solvers
<IMG
WIDTH="119" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img70.png"
ALT="$z_i^k=(A_i^k)^{-1} w_i^k$">;
ALT="$z_i^k=(A_i^k)^{-1} w_i^k$">;
</LI>
<LI>the prolongation and the sum of the previous vectors,
i.e. <!-- MATH
<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
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="$z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$">.
</LI>
</UL><FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">
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
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 [<A
HREF="node29.html#CAI_SARKIS">6</A>].
Direct solvers based on sparse LU factorizations, implemented in the
third-party libraries reported in Section <AHREF="node7.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.
Salvatore Filippone
8 years ago