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@@ -55,46 +55,46 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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Multigrid Background
</H1>␍␍␍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
HREF="node27.html#Briggs2000">2</A>,<A
HREF="node27.html#Stuben_01">27</A>,<A
HREF="node27.html#dd2_96">25</A>] for details.)␍␍MLD2P4 uses a pure algebraic approach, based on the smoothed ␍aggregation algorithm [<A
HREF="node27.html#BREZINA_VANEK">1</A>,<A
HREF="node27.html#VANEK_MANDEL_BREZINA">29</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="node27.html#TUMINARO_TONG">28</A>].␍A brief description of the AMG preconditioners implemented in MLD2P4 is given in ␍Sections&nbsp;<A HREF="node12.html#sec:multilevel">4.1</A>-<A HREF="#sec:smoothers">4.3</A>. For further details the reader␍is referred to [<A
HREF="node27.html#para_04">3</A>,<A
HREF="node27.html#aaecc_07">4</A>,<A
HREF="node27.html#apnum_07">5</A>,<A
HREF="node27.html#MLD2P4_TOMS">9</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.␍␍␍<BR><HR>
HREF="node26.html#Briggs2000">3</A>,<A
HREF="node26.html#Stuben_01">23</A>,<A
HREF="node26.html#dd2_96">21</A>] for details.)␍␍MLD2P4 uses a pure algebraic approach, based on the smoothed ␍aggregation algorithm [<A
HREF="node26.html#BREZINA_VANEK">2</A>,<A
HREF="node26.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="node26.html#TUMINARO_TONG">24</A>].␍A brief description of the AMG preconditioners implemented in MLD2P4 is given in ␍Sections&nbsp;<A HREF="node12.html#sec:multilevel">4.1</A>-<A HREF="#sec:smoothers">4.3</A>. For further details the reader␍is referred to [<A
HREF="node26.html#para_04">4</A>,<A
HREF="node26.html#aaecc_07">5</A>,<A
HREF="node26.html#apnum_07">7</A>,<A
HREF="node26.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.␍␍␍<BR><HR>
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