Parallel Time Integration with Multigrid

We consider optimal-scaling multigrid solvers for the linear systems that arise from the discretization of problems with evolutionary behavior. Typically, solution algorithms for evolution equations are based on a time-marching approach, solving sequentially for one time step after the other. Parall...

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Bibliographic Details
Published in:SIAM journal on scientific computing Vol. 36; no. 6; pp. C635 - C661
Main Authors: Falgout, R. D., Friedhoff, S., Kolev, Tz. V., MacLachlan, S. P., Schroder, J. B.
Format: Journal Article
Language:English
Published: 01.01.2014
Subjects:
Algorithms
Discretization
Evolutionary
Mathematical analysis
Mathematical models
Processors
Reduction
Time marching
ISSN:1064-8275, 1095-7197
Online Access:Get full text
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Summary:We consider optimal-scaling multigrid solvers for the linear systems that arise from the discretization of problems with evolutionary behavior. Typically, solution algorithms for evolution equations are based on a time-marching approach, solving sequentially for one time step after the other. Parallelism in these traditional time-integration techniques is limited to spatial parallelism. However, current trends in computer architectures are leading toward systems with more, but not faster, processors. Therefore, faster compute speeds must come from greater parallelism. One approach to achieving parallelism in time is with multigrid, but extending classical multigrid methods for elliptic operators to this setting is not straightforward. In this paper, we present a nonintrusive, optimal-scaling time-parallel method based on multigrid reduction (MGR). We demonstrate optimality of our multigrid-reduction-in-time algorithm (MGRIT) for solving diffusion equations in two and three space dimensions in numerical experiments. Furthermore, through both parallel performance models and actual parallel numerical results, we show that we can achieve significant speedup in comparison to sequential time marching on modern architectures.
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ISSN:1064-8275
1095-7197
DOI:10.1137/130944230