Analysis and simulation of rare events for SPDEs

Charles-Edouard Bréhier; Maxime Gazeau; Ludovic Goudenège; Mathias Rousset

doi:10.1051/proc/201448017

Geodesic

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Analysis and simulation of rare events for SPDEs

Charles-Edouard Bréhier ^1,² ; Maxime Gazeau ³ ; Ludovic Goudenège ⁴ ; Mathias Rousset ^1,²

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

ESAIM. Proceedings, Tome 48 (2015), pp. 364-384.

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Résumé

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).

DOI : 10.1051/proc/201448017

Affiliations des auteurs :

Charles-Edouard Bréhier ^{1, 2} ; Maxime Gazeau ³ ; Ludovic Goudenège ⁴ ; Mathias Rousset ^{1, 2}

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     title = {Analysis and simulation of rare events for {SPDEs}},
     journal = {ESAIM. Proceedings},
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     doi = {10.1051/proc/201448017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/201448017/}
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Charles-Edouard Bréhier; Maxime Gazeau; Ludovic Goudenège; Mathias Rousset. Analysis and simulation of rare events for SPDEs. ESAIM. Proceedings, Tome 48 (2015), pp. 364-384. doi : 10.1051/proc/201448017. http://geodesic.mathdoc.fr/articles/10.1051/proc/201448017/

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