A Parareal Algorithm with Low-Rank Coarse Solvers
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913178399539200 |
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| author | Gander, Martin J. Ohlberger, Mario Rave, Stephan |
| author_facet | Gander, Martin J. Ohlberger, Mario Rave, Stephan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08873 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Parareal Algorithm with Low-Rank Coarse Solvers Gander, Martin J. Ohlberger, Mario Rave, Stephan Numerical Analysis 65M20, 65M55, 65Y05, 65Y20 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. |
| title | A Parareal Algorithm with Low-Rank Coarse Solvers |
| topic | Numerical Analysis 65M20, 65M55, 65Y05, 65Y20 |
| url | https://arxiv.org/abs/2508.08873 |