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Analysis and simulation of rare events for SPDEs

Abstract : In this work, we consider the numerical estimation of the probability for a stochastic process to hit a set B before reaching another set A. This event is assumed to be rare. We consider reactive trajectories of the stochastic Allen-Cahn partial differential evolution equation (with double well potential) in dimension 1. Reactive trajectories are defined as the probability distribution of the trajectories of a stochastic process, conditioned by the event of hitting B before A. We investigate the use of the so-called Adaptive Multilevel Splitting algorithm in order to estimate the rare event and simulate reactive trajectories. This algorithm uses a reaction coordinate (a real valued function of state space defining level sets), and is based on (i) the selection, among several replicas of the system having hit A before B, of those with maximal reaction coordinate; (ii) iteration of the latter step. We choose for the reaction coordinate the average magnetization, and for B the minimum of the well opposite to the initial condition. We discuss the context, prove that the algorithm has a sense in the usual functional setting, and numerically test the method (estimation of rare event, and transition state sampling).
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Submitted on : Wednesday, June 3, 2020 - 10:16:16 AM
Last modification on : Wednesday, October 20, 2021 - 12:24:01 AM

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Charles-Edouard Bréhier, Maxime Gazeau, Ludovic Goudenège, Mathias Rousset. Analysis and simulation of rare events for SPDEs. ESAIM: Proceedings and Surveys, EDP Sciences, 2015, 48, pp.364-384. ⟨10.1051/proc/201448017⟩. ⟨hal-02746493⟩

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