Standard multigrid algorithms must lead to processor idle time on large-scale parallel computers because the coarsest grids have fewer points than processors. In some cases, this may be considered to be a disadvantage...
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Standard multigrid algorithms must lead to processor idle time on large-scale parallel computers because the coarsest grids have fewer points than processors. In some cases, this may be considered to be a disadvantage. Frederickson and McBryan [multigrid Methods, Marcel Dekker, New York, 1988] show that retaining all points on all grid levels (using all processors) can lead to a "superconvergent" algorithm in that a very good convergence rate is obtained. Has the "parallel superconvergent" multigrid algorithm (PSMG) of Frederickson and McBryan solved the problem of implementing multigrid on a massively parallel single-instruction-multiple-data (SIMD) architecture? How much can be gained by retaining all points on all grid levels, keeping all processors busy? The purpose of this note is to compare the parallel efficiency of the PSMG algorithm to a standard multigrid algorithm. It is shown that the perfect processor utilization and the good convergence rates of the PSMG algorithm do lead to a more efficient algorithm for the special case of one (or fewer) grid points per processor. Normalized computation and communication requirements are given, so that the two types of algorithms can be compared directly.
In a previous paper ["parallel Superconvergent multigrid," in multigrid Methods, Marcel Dekker, New York, 1988] the authors introduced an efficient multiscale PDE solver for massively parallel architectures,...
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In a previous paper ["parallel Superconvergent multigrid," in multigrid Methods, Marcel Dekker, New York, 1988] the authors introduced an efficient multiscale PDE solver for massively parallel architectures, which was called parallel Superconvergent multigrid, or PSMG. In this paper, sharp estimates are derived for the normalized work involved in PSMG solution-the number of parallel arithmetic and communication operations required per digit of error reduction. PSMG is shown to provide fourth-order accurate solutions of Poisson-type equations at convergence rates of .00165 per single relaxation iteration, and with parallel operation counts per grid level of 5.75 communications and 8.62 computations for each digit of error reduction. The authors show that PSMG requires less than one-half as many arithmetic and one-fifth as many communication operations, per digit of error reduction, as a parallel standard multigrid algorithm (RBTRB) presented recently by Decker [SIAM J. Sci. Statist. Comput., 12 (1991), pp. 208-220].
Symmetry and antisymmetry properties of a class of elliptic partial differential equations are exploited to prove when a particular parallel multilevel algorithm is a direct method rather than the usual iterative meth...
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Symmetry and antisymmetry properties of a class of elliptic partial differential equations are exploited to prove when a particular parallel multilevel algorithm is a direct method rather than the usual iterative method. No smoothing is required for this result. Examples are presented, including variable coefficient ones. A connection between the algorithm in this article and domain decomposition is established, even though this algorithm is more general and different. The parallel algorithm is also analyzed when it is iterative and it is shown how to increase processor utilization. Hackbusch’s robust multigrid algorithm [“A new approach to robust multi-grid solvers,” in ICIAM ’87 : Proceedings of the First International Conference on Industrial and Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1988, pp. 111–126] is analyzed for some model problems and it is shown that the parallel algorithm in this article uses much less computer time with at most the same amount of storage.
multigrid methods are distinguished by their optimal (sequential) efficiency and by the fact that all their algorithmical components are fully parallelizable. For this reason, this class of numerical methods is especi...
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multigrid methods are distinguished by their optimal (sequential) efficiency and by the fact that all their algorithmical components are fully parallelizable. For this reason, this class of numerical methods is especially attractive for use on parallel (MIMD, local memory) computers. In this paper, we describe a parallel multigrid solver for steady-state incompressible Navier-Stokes equations on general domains which is currently being developed at the GMD. Due to the geometrical generality of the problem, our approach is based on a non-staggered (nodal-point) finite volume scheme on multi-block boundary fitted grids. The typical instability of non-staggered schemes is overcome by suitably modifying the discrete continuity equation without affecting the overall order of consistency. Starting from the most simple Cartesian case, we discuss several possible multigrid approaches to the general 2D-problem. This motivates the basic design decisions of our multigrid solver in regard to both the discretization and the choice of multigrid components (smoothing schemes). Furthermore, the principal technique of parallelization (grid partitioning) is described as well as some fundamental aspects of the implementation (communication library).
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