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Interpretation of parareal as a two-level additive Schwarz in time preconditioner and its acceleration with GMRES

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Abstract

We describe an interpretation of parareal as a two-level additive Schwarz preconditioner in the time domain. We show that this two-level preconditioner in time is equivalent to parareal and to multigrid reduction in time (MGRIT) with F-relaxation. We also discuss the case when additional fine or coarse propagation steps are applied in the preconditioner. This leads to procedures equivalent to MGRIT with FCF-relaxation and to MGRIT with F(CF)2-relaxation or overlapping parareal. Numerical results show that these variants have faster convergence in some cases. In addition, we also apply a Krylov subspace method, namely GMRES (generalized minimal residual), to accelerate the parareal algorithm. Better convergence is obtained, especially for the advection-reaction-diffusion equation in the case when advection and reaction coefficients are large.

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Acknowledgements

We would like to thank Martin J. Gander for his valuable discussions and the reviewers for constructive comments that helped us improve the manuscript.

Funding

Funding was provided by the French National Research Agency (ANR) Contract ANR-15-CE23-0019 (project CINE-PARA).

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    Authors and Affiliations

    1. Inria, Project-Team ALPINES, 2 Rue Simone IFF, Paris, 75012, France

      Van-Thanh Nguyen & Laura Grigori

    2. CNRS, Laboratoire Jacques-Louis Lions (LJLL), Sorbonne Université and Université de Paris, 4 Place Jussieu, Paris, 75005, France

      Van-Thanh Nguyen & Laura Grigori

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    1. Van-Thanh Nguyen
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    Contributions

    - Describe an interpretation of parareal as a two-level additive Schwarz preconditioner in the time domain so-called SC two-level additive Schwarz in time preconditioner.

    - Show the equivalence between the three methods: SC two-level additive Schwarz in time preconditioner, MGRIT with F-relaxation and parareal.

    - Introduce some variants as SCS, S(CS)2 two-level additive Schwarz in time preconditioner and show their equivalence to MGRIT with FCF-relaxation, and to MGRIT with F(CF)2-relaxation or overlapping parareal.

    - Propose a variant referred to as SCS2 two-level additive Schwarz in time preconditioner which converges faster and efficiently exploits parallel computing.

    - Present the convergence analysis and convergence estimate of SC two-level additive Schwarz in time preconditioner and its variants.

    - Present the computational cost analysis of parareal with GMRES acceleration.

    - Conduct numerical experiments to show the equivalence between parareal and SC two-level additive Schwarz in time preconditioner, the comparison between SC two-level additive Schwarz in time preconditioner and its variants, the acceleration of parareal with GMRES and the computational cost comparison.

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    Correspondence to Van-Thanh Nguyen.

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    Laura Grigori contributed equally to this work.

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    Nguyen, VT., Grigori, L. Interpretation of parareal as a two-level additive Schwarz in time preconditioner and its acceleration with GMRES. Numer Algor 94, 29–72 (2023). https://doi.org/10.1007/s11075-022-01492-8

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