Issue |
ESAIM: Proc.
Volume 48, January 2015
CEMRACS 2013 - Modelling and simulation of complex systems: stochastic and deterministic approaches
|
|
---|---|---|
Page(s) | 364 - 384 | |
DOI | https://doi.org/10.1051/proc/201448017 | |
Published online | 09 March 2015 |
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
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).
© EDP Sciences, SMAI 2015
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