Geodesic
Analysis and simulation of rare events for SPDEs
Charles-Edouard Bréhier1, 2 ; Maxime Gazeau3 ; Ludovic Goudenège4 ; Mathias Rousset1, 2
1Université Paris-Est, CERMICS (ENPC), 6-8-10 Avenue Blaise Pascal, Cité Descartes, F-77455 Marne-la-Vallée, France;
2INRIA Paris-Rocquencourt, Domaine de Voluceau - Rocquencourt, B.P. 105 - 78153 Le Chesnay, France;
3INRIA Lille - Nord Europe, Parc Scientifique de la Haute-Borne, Park Plaza bâtiment A, 40 avenue Halley, 59650 Villeneuve d’Ascq Cedex, France;
4Fédération 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

Voir la notice de l'article

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/}
}
TY  - JOUR
AU  - Charles-Edouard Bréhier
AU  - Maxime Gazeau
AU  - Ludovic Goudenège
AU  - Mathias Rousset
TI  - Analysis and simulation of rare events for SPDEs
JO  - ESAIM. Proceedings
PY  - 2015
SP  - 364
EP  - 384
VL  - 48
UR  - http://geodesic.mathdoc.fr/articles/10.1051/proc/201448017/
DO  - 10.1051/proc/201448017
LA  - en
ID  - EP_2015_48_a17
ER  - 
%0 Journal Article
%A Charles-Edouard Bréhier
%A Maxime Gazeau
%A Ludovic Goudenège
%A Mathias Rousset
%T Analysis and simulation of rare events for SPDEs
%J ESAIM. Proceedings
%D 2015
%P 364-384
%V 48
%U http://geodesic.mathdoc.fr/articles/10.1051/proc/201448017/
%R 10.1051/proc/201448017
%G en
%F EP_2015_48_a17
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

Cité par Sources :