A Low-Rank Multigrid Method for the Stochastic Steady-State Diffusion Problem

Elman, Howard C.; Su, Tengfei

doi:10.1137/17M1125170

Title: A Low-Rank Multigrid Method for the Stochastic Steady-State Diffusion Problem

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Abstract

We study a multigrid method for solving large linear systems of equations with tensor product structure here. Such systems are obtained from stochastic finite element discretization of stochastic partial differential equations such as the steady-state diffusion problem with random coefficients. When the variance in the problem is not too large, the solution can be well approximated by a low-rank object. In the proposed multigrid algorithm, the matrix iterates are truncated to low rank to reduce memory requirements and computational effort. The technique is proved convergent with an analytic error bound. Numerical experiments show its effectiveness in solving the Galerkin systems compared to the original multigrid solver, especially when the number of degrees of freedom associated with the spatial discretization is large.

Authors:

Elman, Howard C. ^[1]; Su, Tengfei ^[1]

Univ. of Maryland, College Park, MD (United States)

Publication Date:: Thu Mar 15 00:00:00 EDT 2018

Research Org.:: Univ. of Maryland, College Park, MD (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF)

OSTI Identifier:: 1598337

Grant/Contract Number:: SC0009301; DMS-1418754

Resource Type:: Accepted Manuscript

Journal Name:: SIAM Journal on Matrix Analysis and Applications

Additional Journal Information:: Journal Volume: 39; Journal Issue: 1; Journal ID: ISSN 0895-4798

Publisher:: SIAM

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; stochastic finite element method; multigrid; low-rank approximation

Citation Formats


                    Elman, Howard C., and Su, Tengfei. A Low-Rank Multigrid Method for the Stochastic Steady-State Diffusion Problem.  United States: N. p., 2018. 
Web.  doi:10.1137/17M1125170.

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                    Elman, Howard C., & Su, Tengfei. A Low-Rank Multigrid Method for the Stochastic Steady-State Diffusion Problem.  United States.  https://doi.org/10.1137/17M1125170

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                    Elman, Howard C., and Su, Tengfei. Thu .  
"A Low-Rank Multigrid Method for the Stochastic Steady-State Diffusion Problem".  United States.  https://doi.org/10.1137/17M1125170.  https://www.osti.gov/servlets/purl/1598337.

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@article{osti_1598337,

  title        = {A Low-Rank Multigrid Method for the Stochastic Steady-State Diffusion Problem},

  author       = {Elman, Howard C. and Su, Tengfei},

  abstractNote = {We study a multigrid method for solving large linear systems of equations with tensor product structure here. Such systems are obtained from stochastic finite element discretization of stochastic partial differential equations such as the steady-state diffusion problem with random coefficients. When the variance in the problem is not too large, the solution can be well approximated by a low-rank object. In the proposed multigrid algorithm, the matrix iterates are truncated to low rank to reduce memory requirements and computational effort. The technique is proved convergent with an analytic error bound. Numerical experiments show its effectiveness in solving the Galerkin systems compared to the original multigrid solver, especially when the number of degrees of freedom associated with the spatial discretization is large.},

  doi          = {10.1137/17M1125170},

  journal      = {SIAM Journal on Matrix Analysis and Applications},

  number       = 1,

  volume       = 39,

  place        = {United States},

  year         = {Thu Mar 15 00:00:00 EDT 2018},

  month        = {Thu Mar 15 00:00:00 EDT 2018}

}

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