A hybrid parareal Monte Carlo algorithm for parabolic problems

Jad Dabaghi (1) , Yvon Maday (2) , Andrea Zoia (3, 4)

1 ESILV - École Supérieure d'Ingénierie Léonard de Vinci
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
3 CEA - Commissariat à l'énergie atomique et aux énergies alternatives
4 Université Paris-Saclay

In this work, we propose a hybrid Monte Carlo/deterministic ``parareal-in-time'' approach devoted to accelerating Monte Carlo simulations over massively parallel computing environments for the simulation of time-dependent problems. This parareal approach iterates on two different solvers: a low-cost “coarse” solver based on a very cheap deterministic Galerkin scheme and a “fine” solver based on a high-fidelity Monte Carlo resolution. In a set of benchmark numerical experiments based on a toy model concerning the time-dependent diffusion equation, we compare our hybrid parareal strategy with a standard full Monte Carlo solution. In particular, we show that for a large number of processors, our hybrid strategy significantly reduces the computational time of the simulation while preserving its accuracy. The convergence properties of the proposed Monte Carlo/deterministic parareal strategy are also discussed.

Mots clés: Parareal-in-time algorithm; Time-dependent problems; Predictor-corrector; Galerkin schemes; Monte Carlo method

Domaines: Analyse numérique [math.NA]

Jad Dabaghi, Yvon Maday, Andrea Zoia. A hybrid parareal Monte Carlo algorithm for parabolic problems. Journal of Computational and Applied Mathematics, 2023, 420, ⟨10.1016/j.cam.2022.114800⟩. ⟨hal-03143554v5⟩