Analysis and simulation of rare events for SPDEs^*

¹ Université Paris-Est, CERMICS (ENPC), 6-8-10 Avenue Blaise Pascal, Cité Descartes, F-77455 Marne-la-Vallée, France ;
e-mail: brehierc@cermics.enpc.fr
² INRIA Paris-Rocquencourt, Domaine de Voluceau - Rocquencourt, B.P. 105 - 78153 Le Chesnay, France ;
e-mail: mathias.rousset@inria.fr
³ INRIA Lille - Nord Europe, Parc Scientifique de la Haute-Borne, Park Plaza bâtiment A, 40 avenue Halley, 59650 Villeneuve d’Ascq Cedex, France ;
e-mail: maxime.gazeau@inria.fr
⁴ Fédération de Mathématiques de l’École Centrale Paris, CNRS, Grande voie des vignes, 92295 Châtenay-Malabry, France ;
e-mail: goudenege@math.cnrs.fr

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).