Analysis and simulation of rare events for SPDE

Charles-Edouard Bréhier (1) , Maxime Gazeau (2) , Ludovic Goudenège (3) , Mathias Rousset (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 SIMPAF - SImulations and Modeling for PArticles and Fluids
3 FR3487 - Fédération de Mathématiques de l'Ecole Centrale Paris

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 \emph{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).

Domaines: Equations aux dérivées partielles [math.AP]

Charles-Edouard Bréhier, Maxime Gazeau, Ludovic Goudenège, Mathias Rousset. Analysis and simulation of rare events for SPDE. 2014. ⟨hal-00921680⟩