Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations

Yoshihito Kazashi, Fabio Nobile, Eva Vidličková

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8 Citations (Scopus)
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Abstract

We consider the Dynamical Low Rank (DLR) approximation of random parabolic equations and propose a class of fully discrete numerical schemes. Similarly to the continuous DLR approximation, our schemes are shown to satisfy a discrete variational formulation. By exploiting this property, we establish stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a “parabolic” type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in certain cases, the semi-implicit scheme can be unconditionally stable if the randomness in the system is sufficiently small. Furthermore, we show that these schemes can be interpreted as projector-splitting integrators and are strongly related to the scheme proposed in [29, 30], to which our stability analysis applies as well. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions.
Original languageEnglish
Pages (from-to)973–1024
Number of pages52
JournalNumerische Mathematik
Volume149
DOIs
Publication statusPublished - 17 Nov 2021

Keywords

  • stability properties
  • random parabolic equations
  • dynamical low rank approximation
  • projector-splitting scheme
  • fully discrete numerical schemes
  • implicit scheme
  • semi-implicit scheme
  • numerical results

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