An SVD-approach to Jacobi-Davidson solution of nonlinear Helmholtz eigenvalue problems

Mikhail A. Bochev, G.L.G. Sleijpen, A. Sopaheluwakan

    Research output: Contribution to journalArticleAcademicpeer-review

    11 Citations (Scopus)

    Abstract

    Numerical solution of the Helmholtz equation in an infinite domain often involves restriction of the domain to a bounded computational window where a numerical solution method is applied. On the boundary of the computational window artificial transparent boundary conditions are posed, for example, widely used perfectly matched layers (PMLs) or absorbing boundary conditions (ABCs). Recently proposed transparent-influx boundary conditions (TIBCs) resolve a number of drawbacks typically attributed to PMLs and ABCs, such as introduction of spurious solutions and the inability to have a tight computational window. Unlike the PMLs or ABCs, the TIBCs lead to a nonlinear dependence of the boundary integral operator on the frequency. Thus, a nonlinear Helmholtz eigenvalue problem arises. This paper presents an approach for solving such nonlinear eigenproblems which is based on a truncated singular value decomposition (SVD) polynomial approximation of the nonlinearity and subsequent solution of the obtained approximate polynomial eigenproblem with the Jacobi-Davidson method. The suggested truncated SVD polynomial approximation seems to be of interest on its own. It can be applied in combination with existing eigensolvers to the problems where the nonlinearity is expensive to evaluate or not explicitly given.
    Original languageUndefined
    Article number10.1016/j/laa.2009.03.024
    Pages (from-to)427-440
    Number of pages14
    JournalLinear algebra and its applications
    Volume431
    Issue number3-4
    DOIs
    Publication statusPublished - 2009

    Keywords

    • MSC-65F15
    • METIS-264214
    • Jacobi-Davidson method
    • EWI-16954
    • IR-68836
    • proper orthogonal decomposition (POD)
    • principal component approximation (PCA)
    • transparent boundary conditions
    • Truncated SVD
    • nonlinear eigenvalue problems
    • Helmholtz equation
    • MSC-65F30
    • MSC-35J05

    Cite this

    Bochev, Mikhail A. ; Sleijpen, G.L.G. ; Sopaheluwakan, A. / An SVD-approach to Jacobi-Davidson solution of nonlinear Helmholtz eigenvalue problems. In: Linear algebra and its applications. 2009 ; Vol. 431, No. 3-4. pp. 427-440.
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    abstract = "Numerical solution of the Helmholtz equation in an infinite domain often involves restriction of the domain to a bounded computational window where a numerical solution method is applied. On the boundary of the computational window artificial transparent boundary conditions are posed, for example, widely used perfectly matched layers (PMLs) or absorbing boundary conditions (ABCs). Recently proposed transparent-influx boundary conditions (TIBCs) resolve a number of drawbacks typically attributed to PMLs and ABCs, such as introduction of spurious solutions and the inability to have a tight computational window. Unlike the PMLs or ABCs, the TIBCs lead to a nonlinear dependence of the boundary integral operator on the frequency. Thus, a nonlinear Helmholtz eigenvalue problem arises. This paper presents an approach for solving such nonlinear eigenproblems which is based on a truncated singular value decomposition (SVD) polynomial approximation of the nonlinearity and subsequent solution of the obtained approximate polynomial eigenproblem with the Jacobi-Davidson method. The suggested truncated SVD polynomial approximation seems to be of interest on its own. It can be applied in combination with existing eigensolvers to the problems where the nonlinearity is expensive to evaluate or not explicitly given.",
    keywords = "MSC-65F15, METIS-264214, Jacobi-Davidson method, EWI-16954, IR-68836, proper orthogonal decomposition (POD), principal component approximation (PCA), transparent boundary conditions, Truncated SVD, nonlinear eigenvalue problems, Helmholtz equation, MSC-65F30, MSC-35J05",
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    An SVD-approach to Jacobi-Davidson solution of nonlinear Helmholtz eigenvalue problems. / Bochev, Mikhail A.; Sleijpen, G.L.G.; Sopaheluwakan, A.

    In: Linear algebra and its applications, Vol. 431, No. 3-4, 10.1016/j/laa.2009.03.024, 2009, p. 427-440.

    Research output: Contribution to journalArticleAcademicpeer-review

    TY - JOUR

    T1 - An SVD-approach to Jacobi-Davidson solution of nonlinear Helmholtz eigenvalue problems

    AU - Bochev, Mikhail A.

    AU - Sleijpen, G.L.G.

    AU - Sopaheluwakan, A.

    N1 - Please note different possible spellings of the first author name: "Bochev" or "Botchev".

    PY - 2009

    Y1 - 2009

    N2 - Numerical solution of the Helmholtz equation in an infinite domain often involves restriction of the domain to a bounded computational window where a numerical solution method is applied. On the boundary of the computational window artificial transparent boundary conditions are posed, for example, widely used perfectly matched layers (PMLs) or absorbing boundary conditions (ABCs). Recently proposed transparent-influx boundary conditions (TIBCs) resolve a number of drawbacks typically attributed to PMLs and ABCs, such as introduction of spurious solutions and the inability to have a tight computational window. Unlike the PMLs or ABCs, the TIBCs lead to a nonlinear dependence of the boundary integral operator on the frequency. Thus, a nonlinear Helmholtz eigenvalue problem arises. This paper presents an approach for solving such nonlinear eigenproblems which is based on a truncated singular value decomposition (SVD) polynomial approximation of the nonlinearity and subsequent solution of the obtained approximate polynomial eigenproblem with the Jacobi-Davidson method. The suggested truncated SVD polynomial approximation seems to be of interest on its own. It can be applied in combination with existing eigensolvers to the problems where the nonlinearity is expensive to evaluate or not explicitly given.

    AB - Numerical solution of the Helmholtz equation in an infinite domain often involves restriction of the domain to a bounded computational window where a numerical solution method is applied. On the boundary of the computational window artificial transparent boundary conditions are posed, for example, widely used perfectly matched layers (PMLs) or absorbing boundary conditions (ABCs). Recently proposed transparent-influx boundary conditions (TIBCs) resolve a number of drawbacks typically attributed to PMLs and ABCs, such as introduction of spurious solutions and the inability to have a tight computational window. Unlike the PMLs or ABCs, the TIBCs lead to a nonlinear dependence of the boundary integral operator on the frequency. Thus, a nonlinear Helmholtz eigenvalue problem arises. This paper presents an approach for solving such nonlinear eigenproblems which is based on a truncated singular value decomposition (SVD) polynomial approximation of the nonlinearity and subsequent solution of the obtained approximate polynomial eigenproblem with the Jacobi-Davidson method. The suggested truncated SVD polynomial approximation seems to be of interest on its own. It can be applied in combination with existing eigensolvers to the problems where the nonlinearity is expensive to evaluate or not explicitly given.

    KW - MSC-65F15

    KW - METIS-264214

    KW - Jacobi-Davidson method

    KW - EWI-16954

    KW - IR-68836

    KW - proper orthogonal decomposition (POD)

    KW - principal component approximation (PCA)

    KW - transparent boundary conditions

    KW - Truncated SVD

    KW - nonlinear eigenvalue problems

    KW - Helmholtz equation

    KW - MSC-65F30

    KW - MSC-35J05

    U2 - 10.1016/j.laa.2009.03.024

    DO - 10.1016/j.laa.2009.03.024

    M3 - Article

    VL - 431

    SP - 427

    EP - 440

    JO - Linear algebra and its applications

    JF - Linear algebra and its applications

    SN - 0024-3795

    IS - 3-4

    M1 - 10.1016/j/laa.2009.03.024

    ER -