Geodesic
Analysis and simulation of rare events for SPDEs
Charles-Edouard Bréhier1, 2 ; Maxime Gazeau3 ; Ludovic Goudenège4 ; Mathias Rousset1, 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 Cet article a éte moissonné depuis la source EDP Sciences

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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

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; 
@article{EP_2015_48_a17,
     author = {Charles-Edouard Br\'ehier and Maxime Gazeau and Ludovic Gouden\`ege and Mathias Rousset},
     title = {Analysis and simulation of rare events for {SPDEs}},
     journal = {ESAIM. Proceedings},
     pages = {364--384},
     year = {2015},
     volume = {48},
     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

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