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

Mikhail A. Bochev

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
PublisherDepartment of Applied Mathematics, University of Twente
Number of pages15
StatePublished - 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

Fingerprint

Krylov subspace
Piecewise polynomials
Polynomial approximation
Krylov subspace methods
Linear ordinary differential equations
Exact method
Exponential time
Matrix function
Singular value decomposition
Source terms
Numerical experiment
Unknown
Evaluation
Costs
Demonstrate

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: Department of Applied Mathematics, University of Twente.

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

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

Research output: ProfessionalReport

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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.",
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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, Department of Applied Mathematics, University of Twente, Enschede.

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

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

Research output: ProfessionalReport

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