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Stabilization of parareal algorithms for long-time computation of a class of highly oscillatory Hamiltonian flows using data

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Abstract

Applying parallel-in-time algorithms to multiscale Hamiltonian systems to obtain stable long-time simulations is very challenging. In this paper, we present novel data-driven methods aimed at improving the standard parareal algorithm developed by Lions et al. in 2001, for multiscale Hamiltonian systems. The first method involves constructing a correction operator to improve a given inaccurate coarse solver through solving a Procrustes problem using data collected online along parareal trajectories. The second method involves constructing an efficient, high-fidelity solver by a neural network trained with offline generated data. For the second method, we address the issues of effective data generation and proper loss function design based on the Hamiltonian function. We show proof-of-concept by applying the proposed methods to a Fermi-Pasta-Ulam (FPU) problem. The numerical results demonstrate that the Procrustes parareal method is able to produce solutions that are more stable in energy compared to the standard parareal. The neural network solver can achieve comparable or better runtime performance compared to numerical solvers of similar accuracy. When combined with the standard parareal algorithm, the improved neural network solutions are slightly more stable in energy than the improved numerical coarse solutions.

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Data availability

The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors thank the Texas Advanced Computing Center (TACC) for providing computing resources.

Funding

The authors are partially supported by the National Science Foundation grant DMS-2208504.

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Authors and Affiliations

  1. Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, 78712, USA

    Rui Fang

  2. Department of Mathematics and Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, 78712, USA

    Richard Tsai

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  1. Rui Fang
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Contributions

R.F.: conceptualization, methodology, software, visualization, writing. R.T.: conceptualization, funding acquisition, methodology, project administration, supervision, writing.

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Correspondence to Rui Fang.

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Appendix: A Energy profiles of parareal solutions by various coarse solvers

Appendix: A Energy profiles of parareal solutions by various coarse solvers

Fig. 13
figure 13

Energy profiles of the stiff springs computed from plain parareal solutions by various coarse solvers (\(\Delta t=1.0\), \(T=1000\))

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Fig. 14
figure 14

Energy profiles of the stiff springs computed from Procrustes parareal solutions by various coarse solvers (\(\Delta t=1.0\), \(T=1000\))

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Fang, R., Tsai, R. Stabilization of parareal algorithms for long-time computation of a class of highly oscillatory Hamiltonian flows using data. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01826-8

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