A Parareal Algorithm with Low-Rank Coarse Solvers

Martin J. Gander; Mario Ohlberger; Stephan Rave

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arXiv · 2508.08873

A Parareal Algorithm with Low-Rank Coarse Solvers

Martin J. Gander·Mario Ohlberger·Stephan Rave

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Abstract

We consider a new class of Parareal algorithms, which use ideas from localized reduced basis methods to construct the coarse solver from truncated SVD approximations of the transfer operators mapping initial values for a given time interval to the solution at the end of the interval. By leveraging randomized singular value decompositions, these low-rank approximations are obtained embarrassingly parallel by computing local fine solutions for random initial values. We show a priori and a posteriori error bounds in terms of the computed singular values of the transfer operators. Our numerical experiments demonstrate that our approach can significantly outperform Parareal with single-step coarse solvers. At the same time, it permits to further increase parallelism in Parareal by trading global iterations for a larger number of independent local solves.

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Martin J. Gander, Mario Ohlberger, Stephan Rave. 2025-08-12. A Parareal Algorithm with Low-Rank Coarse Solvers. https://arxiv.org/abs/2508.08873

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