Fourier mode analysis of multigrid methods for partial differential equations with random coefficients

Seynaeve, Bert; Rosseel, Eveline; Nicolai, Bart; Vandewalle, Stefan

doi:10.1016/j.jcp.2006.12.011

Title: Fourier mode analysis of multigrid methods for partial differential equations with random coefficients

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Abstract

Partial differential equations with random coefficients appear for example in reliability problems and uncertainty propagation models. Various approaches exist for computing the stochastic characteristics of the solution of such a differential equation. In this paper, we consider the spectral expansion approach. This method transforms the continuous model into a large discrete algebraic system. We study the convergence properties of iterative methods for solving this discretized system. We consider one-level and multi-level methods. The classical Fourier mode analysis technique is extended towards the stochastic case. This is done by taking the eigenstructure into account of a certain matrix that depends on the random structure of the problem. We show how the convergence properties depend on the particulars of the algorithm, on the discretization parameters and on the stochastic characteristics of the model. Numerical results are added to illustrate some of our theoretical findings.

Authors:

Seynaeve, Bert ^[1]; Rosseel, Eveline ^[1]; Nicolai, Bart ^[2]; Vandewalle, Stefan ^[1]

Katholieke Universiteit Leuven, Department of Computer Science, Celestijnenlaan 200A, B-3001 Leuven (Belgium)
Katholieke Universiteit Leuven, Division of Mechatronics, Biostatistics and Sensors, Willem de Croylaan 42, B-3001 Leuven (Belgium)

Publication Date:: Sun May 20 00:00:00 EDT 2007

OSTI Identifier:: 20991580

Resource Type:: Journal Article

Journal Name:: Journal of Computational Physics

Additional Journal Information:: Journal Volume: 224; Journal Issue: 1; Other Information: DOI: 10.1016/j.jcp.2006.12.011; PII: S0021-9991(06)00621-8; Copyright (c) 2007 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9991

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; CHAOS THEORY; CONVERGENCE; FOURIER ANALYSIS; ITERATIVE METHODS; MATHEMATICAL SOLUTIONS; MATRICES; PARTIAL DIFFERENTIAL EQUATIONS; POLYNOMIALS; RANDOMNESS

Citation Formats


                    Seynaeve, Bert, Rosseel, Eveline, Nicolai, Bart, and Vandewalle, Stefan. Fourier mode analysis of multigrid methods for partial differential equations with random coefficients.  United States: N. p., 2007. 
        Web.  doi:10.1016/j.jcp.2006.12.011.

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                    Seynaeve, Bert, Rosseel, Eveline, Nicolai, Bart, & Vandewalle, Stefan. Fourier mode analysis of multigrid methods for partial differential equations with random coefficients.  United States.  https://doi.org/10.1016/j.jcp.2006.12.011

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                    Seynaeve, Bert, Rosseel, Eveline, Nicolai, Bart, and Vandewalle, Stefan. 2007.  
        "Fourier mode analysis of multigrid methods for partial differential equations with random coefficients".  United States.  https://doi.org/10.1016/j.jcp.2006.12.011.

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@article{osti_20991580,

  title        = {Fourier mode analysis of multigrid methods for partial differential equations with random coefficients},

  author       = {Seynaeve, Bert and Rosseel, Eveline and Nicolai, Bart and Vandewalle, Stefan},

  abstractNote = {Partial differential equations with random coefficients appear for example in reliability problems and uncertainty propagation models. Various approaches exist for computing the stochastic characteristics of the solution of such a differential equation. In this paper, we consider the spectral expansion approach. This method transforms the continuous model into a large discrete algebraic system. We study the convergence properties of iterative methods for solving this discretized system. We consider one-level and multi-level methods. The classical Fourier mode analysis technique is extended towards the stochastic case. This is done by taking the eigenstructure into account of a certain matrix that depends on the random structure of the problem. We show how the convergence properties depend on the particulars of the algorithm, on the discretization parameters and on the stochastic characteristics of the model. Numerical results are added to illustrate some of our theoretical findings.},

  doi          = {10.1016/j.jcp.2006.12.011},

  url          = {https://www.osti.gov/biblio/20991580},
  journal      = {Journal of Computational Physics},

  issn         = {0021-9991},

  number       = 1,

  volume       = 224,

  place        = {United States},

  year         = {Sun May 20 00:00:00 EDT 2007},

  month        = {Sun May 20 00:00:00 EDT 2007}

}

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