Simulating diffusion processes in discontinuous media: a numerical scheme with constant time steps

Antoine Lejay; Géraldine Pichot

hal-00649170, version 2

Simulating diffusion processes in discontinuous media: a numerical scheme with constant time steps

Antoine Lejay (, ) ^1, 2, Géraldine Pichot ()^a^, 3

(02/12/2011)

Résumé : In this article, we propose new Monte Carlo techniques for moving a diffusive particle in a discontinuous media. In this framework, we characterize the stochastic process that governs the positions of the particle. The key tool is the reduction of the process to a Skew Brownian Motion (SBM). In a zone where the coefficients are locally constant on each side of the discontinuity, the new position of the particle after a constant time step is sampled from the exact distribution of the SBM process at the considered time. To do so, we propose two different but equivalent algorithms: a two-steps simulation with a stop at the discontinuity and a one-step direct simulation of the SBM dynamic. Some benchmark tests illustrate their effectiveness.

a – INRIA
1 : Institut Elie Cartan Nancy (IECN)
CNRS : UMR7502 – INRIA – Université Henri Poincaré - Nancy I – Université Nancy II – Institut National Polytechnique de Lorraine (INPL)
2 : TOSCA (INRIA Sophia Antipolis / INRIA Lorraine / IECN)
INRIA – CNRS : UMR7502 – Université Henri Poincaré - Nancy I – Université Nancy II – Institut National Polytechnique de Lorraine (INPL)
3 : SAGE (INRIA - IRISA)
CNRS : UMR6074 – INRIA – Université de Rennes 1

Collaboration : TOSCA ; SAGE

hal-00649170, version 2
http://hal.inria.fr/hal-00649170
oai:hal.inria.fr:hal-00649170
Contributeur : Antoine Lejay <>
Soumis le : Lundi 21 Mai 2012, 14:55:12
Dernière modification le : Jeudi 31 Mai 2012, 14:55:33

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%F hal-00649170, version 2
%0 Unpublished Work
%U http://hal.inria.fr/hal-00649170
%2 MATH:MATH_PR;PHYS:PHYS:PHYS_GEO-PH;SDU:STU:GP;SDE:MCG
%T Simulating diffusion processes in discontinuous media: a numerical scheme with constant time steps
%A Lejay, Antoine
%A Pichot, Géraldine
%+ Institut Elie Cartan Nancy - IECN , TOSCA - INRIA Sophia Antipolis / INRIA Lorraine / IECN , SAGE - INRIA - IRISA
%Z Probabilités et statistiques
%X In this article, we propose new Monte Carlo techniques for moving a diffusive particle in a discontinuous media. In this framework, we characterize the stochastic process that governs the positions of the particle. The key tool is the reduction of the process to a Skew Brownian Motion (SBM). In a zone where the coefficients are locally constant on each side of the discontinuity, the new position of the particle after a constant time step is sampled from the exact distribution of the SBM process at the considered time. To do so, we propose two different but equivalent algorithms: a two-steps simulation with a stop at the discontinuity and a one-step direct simulation of the SBM dynamic. Some benchmark tests illustrate their effectiveness.
%2 Dans cet article, nous proposons une nouvelle méthode de Monte Carlo pour déplacer des particules dans un milieu discontinu. Nous caractérisons le processus stochastique spécifiant la dynamique des particules. Le point clef est la réduction de ce problème à l'utilisation d'un mouvement brownien biaisé. Dans une zone où les coefficients sont localement constants de chaque côté d'une discontinuité, la nouvelle position de la particule après un pas de temps constant est tiré selon la loi exacte du mouvement brownien biaisé au temps considéré. Pour cela, nous proposons deux algorithmes : l'un reposant sur une simuation à deux pas avec un arrêt à l'interface, et l'autre reposant sur une simulation directe. Des cas tests illustrent leurs efficacités.
%2 I.: Computing Methodologies/I.6: SIMULATION AND MODELING/I.6.8: Types of Simulation/I.6.8.6: Monte Carlo, G.: Mathematics of Computing/G.3: PROBABILITY AND STATISTICS/G.3.7: Probabilistic algorithms (including Monte Carlo), G.3.15: Stochastic processes
%Z (AMS 2010) 60J60 ; 65C05 ; 76S05
%G Anglais
%K divergence form operators ; stochastic differential equation ; skew Brownian motion ; Monte Carlo simulation ; Euler scheme ; geophysics ; diffusive media with interfaces
%2 2011-12-02
%2 10271
%Z TOSCA ; SAGE

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