Abstract.

The dynamical low-rank approximation (DLRA) is used to treat high-dimensional problems that arise in such diverse fields as kinetic transport and uncertainty quantification. Even though it is well known that certain spatial and temporal discretizations when combined with the DLRA approach can result in numerical instability, this phenomenon is poorly understood. In this paper we perform an \(L^2\) stability analysis for the corresponding nonlinear equations of motion. This reveals the source of the instability for the projector-splitting integrator [C. Lubich and I. V. Oseledets, BIT, 54 (2014), pp. 171–188] when first discretizing the equations and then applying the DLRA. Based on this we propose a projector-splitting integrator, based on applying DLRA to the continuous system before performing the discretization, that recovers the classic CFL condition. We also show that the unconventional integrator [G. Ceruti and C. Lubich, BIT, 62 (2021), pp. 23–44] has more favorable stability properties. Moreover, we explain why the projector-splitting integrator performs better when approximating higher moments, while the unconventional integrator is generally superior for first order moments. Furthermore, an efficient and stable dynamical low-rank update for the scattering term in kinetic transport is proposed. Numerical experiments for kinetic transport and uncertainty quantification, which confirm the results of the stability analysis, are presented.

Keywords

  1. dynamical low-rank approximation
  2. numerical stability
  3. kinetic equations
  4. uncertainty quantification
  5. projector-splitting integrator
  6. unconventional integrator

MSC codes

  1. 35L65
  2. 37M15
  3. 65C20
  4. 65P40
  5. 65M12

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Acknowledgments.

The authors would like to thank Christian Lubich for his helpful comments and suggestions, which have been important for the presentation as well as a deeper understanding of our stability analysis.

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Information & Authors

Information

Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A1 - A24
ISSN (online): 1095-7197

History

Submitted: 14 September 2021
Accepted: 30 September 2022
Published online: 19 January 2023

Keywords

  1. dynamical low-rank approximation
  2. numerical stability
  3. kinetic equations
  4. uncertainty quantification
  5. projector-splitting integrator
  6. unconventional integrator

MSC codes

  1. 35L65
  2. 37M15
  3. 65C20
  4. 65P40
  5. 65M12

Authors

Affiliations

Karlsruhe Institute of Technology, Karlsruhe, Germany.
University of Innsbruck, Innsbruck, Austria.
Gianluca Ceruti
Universität Tübingen, Tübingen, Germany.

Funding Information

Funding: This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 258734477-SFB 1173.

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