Micro-macro Parareal, from ODEs to SDEs and back again (2401.01798v2)
Abstract: In this paper, 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 to 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 (ODEs). Then, we extend a variant of the micro-macro Parareal algorithm for scalar stochastic differential equations (SDEs) to higher-dimensional SDEs.
- L. Arnold. Stochastic differential equations: theory and applications. Wiley, New York, 1974. ISBN 978-0-471-03359-2.
- Parallel in time algorithms with reduction methods for solving chemical kinetics. Communications in Applied Mathematics and Computational Science, 5(2):241–263, Dec. 2010. ISSN 2157-5452, 1559-3940. doi:10.2140/camcos.2010.5.241.
- I. Bossuyt. micro-macro-parareal-anziam, 2023. URL https://gitlab.kuleuven.be/numa/public/micro-macro-Parareal-ANZIAM.
- Monte-Carlo/Moments micro-macro Parareal method for unimodal and bimodal scalar McKean-Vlasov SDEs, Oct. 2023. URL http://arxiv.org/abs/2310.11365. arXiv:2310.11365 [math.NA, physics, stat].
- M. J. Gander and S. Vandewalle. Analysis of the parareal time-parallel time-integration method. SIAM Journal on Scientific Computing, 29(2):556–578, 2007. ISSN 10648275. doi:10.1137/05064607X.
- A Unified Analysis Framework for Iterative Parallel-in-Time Algorithms. SIAM Journal on Scientific Computing, 45(5):A2275–A2303, Oct. 2023. ISSN 1064-8275, 1095-7197. doi:10.1137/22M1487163.
- P. E. Kloeden and E. Platen. Numerical solution of stochastic differential equations. Number 23 in Applications of mathematics. Springer, Berlin Heidelberg, 1999. ISBN 978-3-540-54062-5 978-3-642-08107-1.
- A micro-macro parareal algorithm: application to singularly perturbed differential equations. SIAM Journal on Scientific Computing, 2013-01, 35(4):p.A1951–A1986, 2013. doi:10.1137/120872681.
- Résolution d’EDP par un schéma en temps “pararéel”. C. R. Acad. Sci. Paris Sér. I Math., 332(7):661–668, 2001. ISSN 0764-4442. doi:10.1016/S0764-4442(00)01793-6. URL https://doi.org/10.1016/S0764-4442(00)01793-6.
- C. Rackauckas and Q. Nie. DifferentialEquations.jl – A Performant and Feature-Rich Ecosystem for Solving Differential Equations in Julia. Journal of Open Research Software, 5(1):15, May 2017. ISSN 2049-9647. doi:10.5334/jors.151.
- A. J. Roberts. Model Emergent Dynamics in Complex Systems. Society for Industrial and Applied Mathematics, Philadelphia, PA, Jan. 2014. ISBN 978-1-61197-355-6 978-1-61197-356-3. doi:10.1137/1.9781611973563.
- R. Rodriguez and H. C. Tuckwell. Statistical properties of stochastic nonlinear dynamical models of single spiking neurons and neural networks. Physical Review E, 54(5):5585–5590, Nov. 1996. ISSN 1063-651X, 1095-3787. doi:10.1103/PhysRevE.54.5585.
- A.-S. Sznitman. Topics in propagation of chaos. In P.-L. Hennequin, editor, Ecole d’Eté de Probabilités de Saint-Flour XIX — 1989, volume 1464, pages 165–251. Springer Berlin Heidelberg, Berlin, Heidelberg, 1991. ISBN 978-3-540-53841-7 978-3-540-46319-1. doi:10.1007/BFb0085169. Series Title: Lecture Notes in Mathematics.
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