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Multigrid methods with space–time concurrency

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Computing and Visualization in Science

Abstract

We consider the comparison of multigrid methods for parabolic partial differential equations that allow space–time concurrency. With current trends in computer architectures leading towards systems with more, but not faster, processors, space–time concurrency is crucial for speeding up time-integration simulations. In contrast, traditional time-integration techniques impose serious limitations on parallel performance due to the sequential nature of the time-stepping approach, allowing spatial concurrency only. This paper considers the three basic options of multigrid algorithms on space–time grids that allow parallelism in space and time: coarsening in space and time, semicoarsening in the spatial dimensions, and semicoarsening in the temporal dimension. We develop parallel software and performance models to study the three methods at scales of up to 16K cores and introduce an extension of one of them for handling multistep time integration. We then discuss advantages and disadvantages of the different approaches and their benefit compared to traditional space-parallel algorithms with sequential time stepping on modern architectures.

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Notes

  1. One possibility to save on memory in the waveform relaxation approach is to subdivide the time interval into a sequence of “windows” that are treated sequentially [43]. However, there is an apparent parallel performance tradeoff with this reduction in storage requirement.

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Author information

Author notes

  1. S. Friedhoff

    Present address: Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, 42097, Wuppertal, Germany

Authors and Affiliations

  1. Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, P. O. Box 808, L-561, Livermore, CA, 94551, USA

    R. D. Falgout, Tz. V. Kolev & J. B. Schroder

  2. Department of Computer Science, KU Leuven, Celestijnenlaan 200a - box 2402, 3001, Leuven, Belgium

    S. Friedhoff & S. Vandewalle

  3. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, Canada

    S. P. MacLachlan

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  1. R. D. Falgout
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  2. S. Friedhoff
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  3. Tz. V. Kolev
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  4. S. P. MacLachlan
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  6. S. Vandewalle
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Correspondence to S. Friedhoff.

Additional information

Communicated by Rolf Krause.

This work performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 (LLNL-JRNL-678572). The work of SF and SPM was partially supported by the National Science Foundation, under Grant DMS-1015370. The work of SPM was partially supported by an NSERC discovery grant. SF and SV acknowledge support from OPTEC (OPTimization in Engineering Center of excellence KU Leuven), which is funded by the KU Leuven Research Council under Grant No. PFV/10/002.

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Falgout, R.D., Friedhoff, S., Kolev, T.V. et al. Multigrid methods with space–time concurrency. Comput. Visual Sci. 18, 123–143 (2017). https://doi.org/10.1007/s00791-017-0283-9

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