Multigrid Reduction in Time for Nonlinear Parabolic Problems: A Case Study

Falgout, R. D.; Manteuffel, T. A.; O'Neill, B.; Schroder, J. B.

doi:10.1137/16m1082330

Title: 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 »« less

Authors:

Falgout, R. D. ^[1]; Manteuffel, T. A. ^[2]; O'Neill, B. ^[2]; Schroder, J. B. ^[1]

Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Univ. of Colorado, Boulder, CO (United States)

Publication Date:: Thu Oct 26 00:00:00 EDT 2017

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.

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                    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

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                    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.

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@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 referenced in this record:

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Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems
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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
journal, February 2018

Hessenthaler, A.; Nordsletten, D.; Röhrle, O.
Numerical Linear Algebra with Applications, Vol. 25, Issue 3
DOI: 10.1002/nla.2155

Performance Evaluation for a PETSc Parallel-in-Time Solver Based on the MGRIT Algorithm
book, December 2018

Mele, Valeria; Romano, Diego; Constantinescu, Emil M.
Lecture Notes in Computer Science
DOI: 10.1007/978-3-030-10549-5_56

Wave propagation characteristics of Parareal
journal, May 2018

Ruprecht, Daniel
Computing and Visualization in Science, Vol. 19, Issue 1-2
DOI: 10.1007/s00791-018-0296-z

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Title: Multigrid Reduction in Time for Nonlinear Parabolic Problems: A Case Study

Abstract

Citation Formats

The time-parallel multigrid method journal, September 1992

A time-parallel multigrid-extrapolation method for parabolic partial differential equations journal, January 1992

A parallel shooting technique for solving dissipative ODE's journal, September 1993

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An Algorithm with Polylog Parallel Complexity for Solving Parabolic Partial Differential Equations journal, May 1995

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Fourier mode analysis of the multigrid waveform relaxation and time-parallel multigrid methods journal, December 1995

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A parallel shooting technique for solving dissipative ODE's
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Least-Squares Finite Element Methods and Algebraic Multigrid Solvers for Linear Hyperbolic PDEs
journal, January 2004

Résolution d'EDP par un schéma en temps «pararéel »
journal, April 2001

On the $p$-Laplacian and $\infty$-Laplacian on Graphs with Applications in Image and Data Processing
journal, January 2015

Parallel Time Integration with Multigrid
journal, January 2014

Analysis of the Parareal Time‐Parallel Time‐Integration Method
journal, January 2007

Efficient Parallel Algorithms for Solving Initial-Boundary Value and Time-Periodic Parabolic Partial Differential Equations
journal, November 1992

Parallel High-Order Integrators
journal, January 2010

Toward an efficient parallel in time method for partial differential equations
journal, January 2012

Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems
journal, January 2016

Parallel methods for integrating ordinary differential equations
journal, December 1964

Multi-level adaptive solutions to boundary-value problems
journal, May 1977

Interweaving PFASST and Parallel Multigrid
journal, January 2015

An Algorithm with Polylog Parallel Complexity for Solving Parabolic Partial Differential Equations
journal, May 1995

Parallel methods for the numerical integration of ordinary differential equations
journal, September 1967

A Space-Time Multigrid Method for Parabolic Partial Differential Equations
journal, July 1995

Parallelization in time through tensor-product space–time solvers
journal, January 2008

A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature
journal, April 2003

Fourier mode analysis of the multigrid waveform relaxation and time-parallel multigrid methods
journal, December 1995

Multigrid Methods for Variational Problems
journal, October 1982

Convergence of the multigrid reduction in time algorithm for the linear elasticity equations: Convergence of the MGRIT algorithm for linear elasticity
journal, February 2018

Performance Evaluation for a PETSc Parallel-in-Time Solver Based on the MGRIT Algorithm
book, December 2018

Wave propagation characteristics of Parareal
journal, May 2018