Multigrid Reduction in Time for Nonlinear Parabolic Problems
Abstract
The need for parallelintime is being driven by changes in computer architectures, where future speedups will be available through greater concurrency, but not faster clock speeds, which are stagnant.This leads to 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 efficient is defined as achieving similar performance when compared to a corresponding linear problem. As our benchmark, we use the pLaplacian, where p = 4 corresponds to a wellknown nonlinear diffusion equation and p = 2 corresponds to our benchmark linear diffusion problem. When considering linear problems and implicit methods, the use of optimal spatial solvers such as spatial multigrid imply that the cost of one time step evaluation is fixed across temporal levels, which have a large variation in time step sizes. This is not the case for nonlinear problems, where the work requiredmore »
 Authors:

 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Univ. of Colorado, Boulder, CO (United States)
 Publication Date:
 Research Org.:
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 1236132
 Report Number(s):
 LLNLTR680499
 DOE Contract Number:
 AC5207NA27344
 Resource Type:
 Technical Report
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE
Citation Formats
Falgout, R. D., Manteuffel, T. A., O'Neill, B., and Schroder, J. B. Multigrid Reduction in Time for Nonlinear Parabolic Problems. United States: N. p., 2016.
Web. doi:10.2172/1236132.
Falgout, R. D., Manteuffel, T. A., O'Neill, B., & Schroder, J. B. Multigrid Reduction in Time for Nonlinear Parabolic Problems. United States. doi:10.2172/1236132.
Falgout, R. D., Manteuffel, T. A., O'Neill, B., and Schroder, J. B. Mon .
"Multigrid Reduction in Time for Nonlinear Parabolic Problems". United States. doi:10.2172/1236132. https://www.osti.gov/servlets/purl/1236132.
@article{osti_1236132,
title = {Multigrid Reduction in Time for Nonlinear Parabolic Problems},
author = {Falgout, R. D. and Manteuffel, T. A. and O'Neill, B. and Schroder, J. B.},
abstractNote = {The need for parallelintime is being driven by changes in computer architectures, where future speedups will be available through greater concurrency, but not faster clock speeds, which are stagnant.This leads to 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 efficient is defined as achieving similar performance when compared to a corresponding linear problem. As our benchmark, we use the pLaplacian, where p = 4 corresponds to a wellknown nonlinear diffusion equation and p = 2 corresponds to our benchmark linear diffusion problem. When considering linear problems and implicit methods, the use of optimal spatial solvers such as spatial multigrid imply that the cost of one time step evaluation is fixed across temporal levels, which have a large variation in time step sizes. This is not the case for nonlinear problems, where the work required increases dramatically on coarser time grids, where relatively large time steps lead to worse conditioned nonlinear solves and increased nonlinear iteration counts per time step evaluation. This is the key difficulty explored by this paper. We show that by using a variety of strategies, most importantly, spatial coarsening and an alternate initial guess to the nonlinear timestep solver, we can reduce the work per time step evaluation over all temporal levels to a range similar with the corresponding linear problem. This allows for parallel scaling behavior comparable to the corresponding linear problem.},
doi = {10.2172/1236132},
journal = {},
number = ,
volume = ,
place = {United States},
year = {2016},
month = {1}
}