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103 lines
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<H1><A NAME="SECTION00060000000000000000"></A><A NAME="sec:background"></A>
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<BR>
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Multigrid Background
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</H1>
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Multigrid preconditioners, coupled with Krylov iterative
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solvers, are widely used in the parallel solution of large and sparse linear systems,
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because of their optimality in the solution of linear systems arising from the
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discretization of scalar elliptic Partial Differential Equations (PDEs) on regular grids.
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Optimality, also known as algorithmic scalability, is the property
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of having a computational cost per iteration that depends linearly on
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the problem size, and a convergence rate that is independent of the problem size.
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Multigrid preconditioners are based on a recursive application of a two-grid process
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consisting of smoother iterations and a coarse-space (or coarse-level) correction.
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The smoothers may be either basic iterative methods, such as the Jacobi and Gauss-Seidel ones,
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or more complex subspace-correction methods, such as the Schwarz ones.
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The coarse-space correction consists of solving, in an appropriately chosen
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coarse space, the residual equation associated with the approximate solution computed
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by the smoother, and of using the solution of this equation to correct the
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previous approximation. The transfer of information between the original
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(fine) space and the coarse one is performed by using suitable restriction and
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prolongation operators. The construction of the coarse space and the corresponding
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transfer operators is carried out by applying a so-called coarsening algorithm to the system
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matrix. Two main approaches can be used to perform coarsening: the geometric approach,
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which exploits the knowledge of some physical grid associated with the matrix
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and requires the user to define transfer operators from the fine
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to the coarse level and vice versa, and the algebraic approach, which builds
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the coarse-space correction and the associate transfer operators using only matrix
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information. The first approach may be difficult when the system comes from
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discretizations on complex geometries;
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furthermore, ad hoc one-level smoothers may be required to get an efficient
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interplay between fine and coarse levels, e.g., when matrices with highly varying coefficients
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are considered. The second approach performs a fully automatic coarsening and enforces the
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interplay between fine and coarse level by suitably choosing the coarse space and
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the coarse-to-fine interpolation (see, e.g., [<A
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HREF="node27.html#Briggs2000">2</A>,<A
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HREF="node27.html#Stuben_01">27</A>,<A
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HREF="node27.html#dd2_96">25</A>] for details.)
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MLD2P4 uses a pure algebraic approach, based on the smoothed
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aggregation algorithm [<A
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HREF="node27.html#BREZINA_VANEK">1</A>,<A
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HREF="node27.html#VANEK_MANDEL_BREZINA">29</A>],
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for building the sequence of coarse matrices and transfer operators,
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starting from the original one.
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A decoupled version of this algorithm is implemented, where the smoothed
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aggregation is applied locally to each submatrix [<A
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HREF="node27.html#TUMINARO_TONG">28</A>].
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A brief description of the AMG preconditioners implemented in MLD2P4 is given in
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Sections <A HREF="node12.html#sec:multilevel">4.1</A>-<A HREF="#sec:smoothers">4.3</A>. For further details the reader
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is referred to [<A
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HREF="node27.html#para_04">3</A>,<A
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HREF="node27.html#aaecc_07">4</A>,<A
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HREF="node27.html#apnum_07">5</A>,<A
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HREF="node27.html#MLD2P4_TOMS">9</A>].
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We note that optimal multigrid preconditioners do not necessarily correspond
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to minimum execution times in a parallel setting. Indeed, to obtain effective parallel
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multigrid preconditioners, a tradeoff between the optimality and the cost of building and
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applying the smoothers and the coarse-space corrections must be achieved. Effective
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parallel preconditioners require algorithmic scalability to be coupled with implementation
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scalability, i.e., a computational cost per iteration which remains (almost) constant as
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the number of parallel processors increases.
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<BR><HR>
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<LI><A NAME="tex2html211"
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HREF="node12.html">AMG preconditioners</A>
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