Parallelism in multigrid methods: How much is too much? |
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Authors: | Lesley R Matheson Robert E Tarjan |
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Affiliation: | (1) NEC Research Institute, 4 Independence Way, 08540 Princeton, New Jersey;(2) Department of Computer Science, Princeton University, 08544 Princeton, New Jersey |
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Abstract: | Multigrid methods are powerful techniques to accelerate the solution of computationally-intensive problems arising in a broad
range of applications. Used in conjunction with iterative processes for solving partial differential equations, multigrid
methods speed up iterative methods by moving the computation from the original mesh covering the problem domain through a
series of coarser meshes. But this hierarchical structure leaves domain-parallel versions of the standard multigrid algorithms
with a deficiency of parallelism on coarser grids. To compensate, several parallel multigrid strategies with more parallelism,
but also more work, have been designed. We examine these parallel strategies and compare them to simpler standard algorithms
to try to determine which techniques are more efficient and practical. We consider three parallel multigrid strategies: (1)
domain-parallel versions of the standard V-cycle and F-cycle algorithms; (2) a multiple coarse grid algorithm, proposed by
Fredrickson and McBryan, which generates several coarse grids for each fine grid; and (3) two Rosendale algorithm, which allow
computation on all grids simultaneously. We study an elliptic model problem on simple domains, discretized with finite difference
techniques on block-structured meshes in two or three dimensions with up to 106 or 109 points, respectively. We analyze performance using three models of parallel computation: the PRAM and two bridging models.
The bridging models reflect the salient characteristics of two kinds of parallel computers: SIMD fine-grain computers, which
contain a large number of small (bitserial) processors, and SPMD medium-grain computers, which have a more modest number of
powerful (single chip) processors. Our analysis suggests that the standard algorithms are substantially more efficient than
algorithms utilizing either parallel strategy. Both parallel strategies need too much extra work to compensate for their extra
parallelism. They require a highly impractical number of processors to be competitive with simpler, standard algorithms. The
analysis also suggests that the F-cycle, with the appropriate optimization techniques, is more efficient than the V-cycle
under a broad range of problem, implementation, and machine characteristics, despite the fact that it exhibits even less parallelism
than the V-cycle.
Research at Princeton University partially supported by the National Science Foundation, Grant No. CCR-8920505, and the Office
of Naval Research, Contract No. N0014-91-J-1463. |
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Keywords: | Parallel multigrid methods parallel computers models of parallel computation |
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