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MICRO-MACRO PARAREAL, FROM ORDINARY DIFFERENTIAL EQUATIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS AND BACK AGAIN

Part of: Stochastic analysis Numerical analysis: Ordinary differential equations Probabilistic methods, simulation and stochastic differential equations Asymptotic theory Theory of computing

Published online by Cambridge University Press:  17 March 2025

IGNACE BOSSUYT Open the ORCID record for IGNACE BOSSUYT [Opens in a new window] ,
STEFAN VANDEWALLE Open the ORCID record for STEFAN VANDEWALLE [Opens in a new window]  and
GIOVANNI SAMAEY Open the ORCID record for GIOVANNI SAMAEY [Opens in a new window]
IGNACE BOSSUYT*
Affiliation:
Department of Computer Science, KU Leuven, Celestijnenlaan 200 A, Box 2402, 3001 Leuven, Belgium; e-mail: stefan.vandewalle@kuleuven.be, giovanni.samaey@kuleuven.be
STEFAN VANDEWALLE
Affiliation:
Department of Computer Science, KU Leuven, Celestijnenlaan 200 A, Box 2402, 3001 Leuven, Belgium; e-mail: stefan.vandewalle@kuleuven.be, giovanni.samaey@kuleuven.be
GIOVANNI SAMAEY
Affiliation:
Department of Computer Science, KU Leuven, Celestijnenlaan 200 A, Box 2402, 3001 Leuven, Belgium; e-mail: stefan.vandewalle@kuleuven.be, giovanni.samaey@kuleuven.be
*
e-mail: ignace.bossuyt1@kuleuven.be
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Abstract

We are concerned with the micro-macro Parareal algorithm for the simulation of initial-value problems. In this algorithm, a coarse (fast) solver is applied sequentially over the time domain and a fine (time-consuming) solver is applied as a corrector in parallel over smaller chunks of the time interval. Moreover, the coarse solver acts on a reduced state variable, which is coupled with the fine state variable through appropriate coupling operators. We first provide a contribution to the convergence analysis of the micro-macro Parareal method for multiscale linear ordinary differential equations. Then, we extend a variant of the micro-macro Parareal algorithm for scalar stochastic differential equations (SDEs) to higher-dimensional SDEs.

Keywords

parallel-in-timePararealmultiscaleMcKean–Vlasov SDEmicro-macromoment modelreduced model

MSC classification

Primary: 65L11: Singularly perturbed problems
Secondary: 34E13: Multiple scale methods 65C30: Stochastic differential and integral equations 68Q10: Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.) 65C35: Stochastic particle methods 60H35: Computational methods for stochastic equations
Type
Research Article
Information
The ANZIAM Journal , Volume 67 , 2025 , e13
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

Arnold, L., Stochastic differential equations: theory and applications (Wiley, New York, 1974).Google Scholar
Blouza, A., Boudin, L. and Kaber, S. M., “Parallel in time algorithms with reduction methods for solving chemical kinetics”, Commun. Appl. Math. Comput. Sci. 5 (2010) 241–263; doi:10.2140/camcos.2010.5.241.CrossRefGoogle Scholar
Bossuyt, I., Git repository with the code accompanying the paper: micro-macro-Parareal-ANZIAM (2023); https://gitlab.kuleuven.be/numa/public/micro-macro-Parareal-ANZIAM.Google Scholar
Bossuyt, I., Vandewalle, S. and Samaey, G., “Monte-Carlo/Moments micro-macro Parareal method for unimodal and bimodal scalar McKean-Vlasov SDEs”, Preprint, 2024, arXiv:2310.11365 [math.NA, physics, stat].Google Scholar
Gander, M. J., Lunet, T., Ruprecht, D. and Speck, R., “A unified analysis framework for iterative parallel-in-time algorithms”, SIAM J. Sci. Comput. 45 (2023) A2275–A2303; doi:10.1137/22M1487163.CrossRefGoogle Scholar
Gander, M. J. and Vandewalle, S., “Analysis of the parareal time-parallel time-integration method”, SIAM J. Sci. Comput. 29 (2007) 556–578; doi:10.1137/05064607X.CrossRefGoogle Scholar
Kloeden, P. E. and Platen, E., Numerical solution of stochastic differential equations, Volume 23 of Appl. Math., (Springer, Berlin–Heidelberg, 1999), doi:10.1007/978-3-662-12616-5.Google Scholar
Legoll, F., Lelièvre, T. and Samaey, G., “A micro-macro parareal algorithm: application to singularly perturbed differential equations”, SIAM J. Sci. Comput. 35 (2013) A1951–A1986; doi:10.1137/120872681.CrossRefGoogle Scholar
Lions, J.-L., Maday, Y. and Turinici, G., “Résolution d’EDP par un schéma en temps “pararéel””, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 661–668; doi:10.1016/S0764-4442(00)01793-6.CrossRefGoogle Scholar
Rackauckas, C. and Nie, Q., “DifferentialEquations.jl – a performant and feature-rich ecosystem for solving differential equations in Julia”, J. Open Res. Softw. 5 (2017) Article ID: 15; doi:10.5334/jors.151.CrossRefGoogle Scholar
Roberts, A. J., Model emergent dynamics in complex systems (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2014); doi:10.1137/1.9781611973563.CrossRefGoogle Scholar
Rodriguez, R. and Tuckwell, H. C., “Statistical properties of stochastic nonlinear dynamical models of single spiking neurons and neural networks”, Phys. Rev. E 54 (1996) 5585–5590; doi:10.1103/PhysRevE.54.5585.CrossRefGoogle ScholarPubMed
Sznitman, A.-S., “Topics in propagation of chaos”, in: Ecole d’Eté de Probabilités de Saint-Flour XIX — 1989, Volume 1464 of Lecture Notes in Math. Ser. (ed. Hennequin, P.-L.) (Springer, Berlin–Heidelberg, 1991) 165–251; doi:10.1007/BFb0085169.CrossRefGoogle Scholar
Van Kampen, N., “Elimination of fast variables”, Phys. Rep. 124 (1985) 69–160; doi:10.1016/0370-1573(85)90002-X.CrossRefGoogle Scholar