Multigrid Reduction in Time for Nonlinear Parabolic Problems: A Case Study
Abstract
The need for parallelism in the time dimension is being driven by changes in computer architectures, where performance increases are now provided through greater concurrency, not faster clock speeds. This creates a bottleneck for sequential time marching schemes because they lack parallelism in the time dimension. Multigrid reduction in time (MGRIT) is an iterative procedure that allows for temporal parallelism by utilizing multigrid reduction techniques and a multilevel hierarchy of coarse time grids. MGRIT has been shown to be effective for linear problems, with speedups of up to 50 times. The goal of this work is the efficient solution of nonlinear problems with MGRIT, where efficiency is defined as achieving similar performance when compared to an equivalent linear problem. The benchmark nonlinear problem is the p-Laplacian, where p=4 corresponds to a well-known nonlinear diffusion equation and p=2 corresponds to the standard linear diffusion operator, our benchmark linear problem. The key difficulty encountered is that the nonlinear time-step solver becomes progressively more expensive on coarser time levels as the time-step size increases. To overcome such difficulties, multigrid research has historically targeted an accumulated body of experience regarding how to choose an appropriate solver for a specific problem type. To that end,more »
- Authors:
-
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Univ. of Colorado, Boulder, CO (United States)
- Publication Date:
- Research Org.:
- Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)
- OSTI Identifier:
- 1808764
- Report Number(s):
- LLNL-JRNL-692258
Journal ID: ISSN 1064-8275; 820554
- Grant/Contract Number:
- AC52-07NA27344; FC02-03ER25574; NA0002376
- Resource Type:
- Accepted Manuscript
- Journal Name:
- SIAM Journal on Scientific Computing
- Additional Journal Information:
- Journal Volume: 39; Journal Issue: 5; Journal ID: ISSN 1064-8275
- Publisher:
- Society for Industrial and Applied Mathematics (SIAM)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; multigrid; multigrid-in-time; parabolic problems; nonlinear; reduction-based multigrid; parareal; high performance computing
Citation Formats
Falgout, R. D., Manteuffel, T. A., O'Neill, B., and Schroder, J. B. Multigrid Reduction in Time for Nonlinear Parabolic Problems: A Case Study. United States: N. p., 2017.
Web. doi:10.1137/16m1082330.
Falgout, R. D., Manteuffel, T. A., O'Neill, B., & Schroder, J. B. Multigrid Reduction in Time for Nonlinear Parabolic Problems: A Case Study. United States. https://doi.org/10.1137/16m1082330
Falgout, R. D., Manteuffel, T. A., O'Neill, B., and Schroder, J. B. Thu .
"Multigrid Reduction in Time for Nonlinear Parabolic Problems: A Case Study". United States. https://doi.org/10.1137/16m1082330. https://www.osti.gov/servlets/purl/1808764.
@article{osti_1808764,
title = {Multigrid Reduction in Time for Nonlinear Parabolic Problems: A Case Study},
author = {Falgout, R. D. and Manteuffel, T. A. and O'Neill, B. and Schroder, J. B.},
abstractNote = {The need for parallelism in the time dimension is being driven by changes in computer architectures, where performance increases are now provided through greater concurrency, not faster clock speeds. This creates a bottleneck for sequential time marching schemes because they lack parallelism in the time dimension. Multigrid reduction in time (MGRIT) is an iterative procedure that allows for temporal parallelism by utilizing multigrid reduction techniques and a multilevel hierarchy of coarse time grids. MGRIT has been shown to be effective for linear problems, with speedups of up to 50 times. The goal of this work is the efficient solution of nonlinear problems with MGRIT, where efficiency is defined as achieving similar performance when compared to an equivalent linear problem. The benchmark nonlinear problem is the p-Laplacian, where p=4 corresponds to a well-known nonlinear diffusion equation and p=2 corresponds to the standard linear diffusion operator, our benchmark linear problem. The key difficulty encountered is that the nonlinear time-step solver becomes progressively more expensive on coarser time levels as the time-step size increases. To overcome such difficulties, multigrid research has historically targeted an accumulated body of experience regarding how to choose an appropriate solver for a specific problem type. To that end, this paper develops a library of MGRIT optimizations and modifications, most important an alternate initial guess for the nonlinear time-step solver and delayed spatial coarsening, that will allow many nonlinear parabolic problems to be solved with parallel scaling behavior comparable to the corresponding linear problem.},
doi = {10.1137/16m1082330},
journal = {SIAM Journal on Scientific Computing},
number = 5,
volume = 39,
place = {United States},
year = {Thu Oct 26 00:00:00 EDT 2017},
month = {Thu Oct 26 00:00:00 EDT 2017}
}
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Works referencing / citing this record:
Convergence of the multigrid reduction in time algorithm for the linear elasticity equations: Convergence of the MGRIT algorithm for linear elasticity
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