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Low-rank Parareal: a low-rank parallel-in-time integrator.

Benjamin Carrel1, Martin J Gander1, Bart Vandereycken1

  • 1Section of Mathematics, University of Geneva, Geneva, Switzerland.

BIT. Numerical Mathematics
|February 9, 2023
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Summary

This study introduces a low-rank Parareal algorithm, a novel time-parallel solver for evolution problems. It leverages dynamical low-rank approximation (DLRA) for efficient and accurate computations, especially for problems admitting good low-rank approximations.

Keywords:
Dynamical low-rank approximationInitial value problemMatrix differential equationParallel algorithm

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Area of Science:

  • Numerical Analysis
  • Computational Science
  • Applied Mathematics

Background:

  • Evolution problems often require computationally intensive time-stepping methods.
  • Dynamical low-rank approximation (DLRA) offers efficient solutions for problems with good low-rank approximations.
  • Existing DLRA integrators' performance depends heavily on the chosen approximation rank.

Purpose of the Study:

  • To develop a time-parallel solver for evolution problems using DLRA.
  • To introduce the low-rank Parareal algorithm by combining Parareal with DLRA.
  • To analyze the performance of this new algorithm on specific problem types.

Main Methods:

  • Application of the Parareal algorithm to evolution problems.
  • Utilization of dynamical low-rank approximation (DLRA) as the time stepper.
  • Development of a novel low-rank Parareal method tailored for DLRA.

Main Results:

  • The proposed low-rank Parareal algorithm provides a time-parallel DLRA solver.
  • The method's cost and accuracy are influenced by the approximation rank.
  • Numerical illustrations confirm the algorithm's effectiveness on affine linear problems.

Conclusions:

  • The low-rank Parareal algorithm is a viable approach for accelerating DLRA-based simulations.
  • This method offers a promising direction for efficient parallel-in-time computation.
  • Further research can explore its application to a broader range of evolution problems.