A block Krylov subspace time-exact solution method for linear ODE systems

Mikhail A. Bochev

    Research output: Book/ReportReportProfessional

    35 Downloads (Pure)

    Abstract

    We propose a time-exact Krylov-subspace-based method for solving linear ODE (ordinary differential equation) systems of the form $y'=-Ay + g(t)$ and $y''=-Ay + g(t)$, where $y(t)$ is the unknown function. The method consists of two stages. The first stage is an accurate piecewise polynomial approximation of the source term $g(t)$, constructed with the help of the truncated SVD (singular value decomposition). The second stage is a special residual-based block Krylov subspace method. The accuracy of the method is only restricted by the accuracy of the piecewise polynomial approximation and by the error of the block Krylov process. Since both errors can, in principle, be made arbitrarily small, this yields, at some costs, a time-exact method. Numerical experiments are presented to demonstrate efficiency of the new method, as compared to an exponential time integrator with Krylov subspace matrix function evaluations.
    Original languageUndefined
    Place of PublicationEnschede
    PublisherUniversity of Twente, Department of Applied Mathematics
    Number of pages15
    Publication statusPublished - Jan 2012

    Publication series

    NameMemorandum / Department of Applied Mathematics
    PublisherUniversity of Twente, Department of Applied Mathematics
    No.1973
    ISSN (Print)1874-4850
    ISSN (Electronic)1874-4850

    Keywords

    • MSC-65F60
    • MSC-65F30
    • MSC-65M22
    • MSC-65M20
    • EWI-21277
    • Unconditionally stable time integration
    • Proper orthogonal decomposition
    • Truncated SVD
    • Exponential residual
    • Matrix exponential
    • Block Krylov subspace methods
    • Exponential time integration
    • IR-79354
    • MSC-65L05
    • METIS-285233

    Cite this

    Bochev, M. A. (2012). A block Krylov subspace time-exact solution method for linear ODE systems. (Memorandum / Department of Applied Mathematics; No. 1973). Enschede: University of Twente, Department of Applied Mathematics.
    Bochev, Mikhail A. / A block Krylov subspace time-exact solution method for linear ODE systems. Enschede : University of Twente, Department of Applied Mathematics, 2012. 15 p. (Memorandum / Department of Applied Mathematics; 1973).
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    abstract = "We propose a time-exact Krylov-subspace-based method for solving linear ODE (ordinary differential equation) systems of the form $y'=-Ay + g(t)$ and $y''=-Ay + g(t)$, where $y(t)$ is the unknown function. The method consists of two stages. The first stage is an accurate piecewise polynomial approximation of the source term $g(t)$, constructed with the help of the truncated SVD (singular value decomposition). The second stage is a special residual-based block Krylov subspace method. The accuracy of the method is only restricted by the accuracy of the piecewise polynomial approximation and by the error of the block Krylov process. Since both errors can, in principle, be made arbitrarily small, this yields, at some costs, a time-exact method. Numerical experiments are presented to demonstrate efficiency of the new method, as compared to an exponential time integrator with Krylov subspace matrix function evaluations.",
    keywords = "MSC-65F60, MSC-65F30, MSC-65M22, MSC-65M20, EWI-21277, Unconditionally stable time integration, Proper orthogonal decomposition, Truncated SVD, Exponential residual, Matrix exponential, Block Krylov subspace methods, Exponential time integration, IR-79354, MSC-65L05, METIS-285233",
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    year = "2012",
    month = "1",
    language = "Undefined",
    series = "Memorandum / Department of Applied Mathematics",
    publisher = "University of Twente, Department of Applied Mathematics",
    number = "1973",

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    Bochev, MA 2012, A block Krylov subspace time-exact solution method for linear ODE systems. Memorandum / Department of Applied Mathematics, no. 1973, University of Twente, Department of Applied Mathematics, Enschede.

    A block Krylov subspace time-exact solution method for linear ODE systems. / Bochev, Mikhail A.

    Enschede : University of Twente, Department of Applied Mathematics, 2012. 15 p. (Memorandum / Department of Applied Mathematics; No. 1973).

    Research output: Book/ReportReportProfessional

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    T1 - A block Krylov subspace time-exact solution method for linear ODE systems

    AU - Bochev, Mikhail A.

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    N2 - We propose a time-exact Krylov-subspace-based method for solving linear ODE (ordinary differential equation) systems of the form $y'=-Ay + g(t)$ and $y''=-Ay + g(t)$, where $y(t)$ is the unknown function. The method consists of two stages. The first stage is an accurate piecewise polynomial approximation of the source term $g(t)$, constructed with the help of the truncated SVD (singular value decomposition). The second stage is a special residual-based block Krylov subspace method. The accuracy of the method is only restricted by the accuracy of the piecewise polynomial approximation and by the error of the block Krylov process. Since both errors can, in principle, be made arbitrarily small, this yields, at some costs, a time-exact method. Numerical experiments are presented to demonstrate efficiency of the new method, as compared to an exponential time integrator with Krylov subspace matrix function evaluations.

    AB - We propose a time-exact Krylov-subspace-based method for solving linear ODE (ordinary differential equation) systems of the form $y'=-Ay + g(t)$ and $y''=-Ay + g(t)$, where $y(t)$ is the unknown function. The method consists of two stages. The first stage is an accurate piecewise polynomial approximation of the source term $g(t)$, constructed with the help of the truncated SVD (singular value decomposition). The second stage is a special residual-based block Krylov subspace method. The accuracy of the method is only restricted by the accuracy of the piecewise polynomial approximation and by the error of the block Krylov process. Since both errors can, in principle, be made arbitrarily small, this yields, at some costs, a time-exact method. Numerical experiments are presented to demonstrate efficiency of the new method, as compared to an exponential time integrator with Krylov subspace matrix function evaluations.

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    KW - MSC-65F30

    KW - MSC-65M22

    KW - MSC-65M20

    KW - EWI-21277

    KW - Unconditionally stable time integration

    KW - Proper orthogonal decomposition

    KW - Truncated SVD

    KW - Exponential residual

    KW - Matrix exponential

    KW - Block Krylov subspace methods

    KW - Exponential time integration

    KW - IR-79354

    KW - MSC-65L05

    KW - METIS-285233

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    BT - A block Krylov subspace time-exact solution method for linear ODE systems

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    Bochev MA. A block Krylov subspace time-exact solution method for linear ODE systems. Enschede: University of Twente, Department of Applied Mathematics, 2012. 15 p. (Memorandum / Department of Applied Mathematics; 1973).