Motivated by the development of efficient Monte Carlo methods for PDE models in molecular dynamics, we establish a new probabilistic interpretation of a family of divergence form operators with discontinuous coefficients at the interface of two open subsets of . This family of operators includes the case of the linearized Poisson-Boltzmann equation used to compute the electrostatic free energy of a molecule. More precisely, we explicitly construct a Markov process whose infinitesimal generator belongs to this family, as the solution of a SDE including a non standard local time term related to the interface of discontinuity. We then prove an extended Feynman-Kac formula for the Poisson-Boltzmann equation. This formula allows us to justify various probabilistic numerical methods to approximate the free energy of a molecule. We analyse the convergence rate of these simulation procedures and numerically compare them on idealized molecules models.
@article{M2AN_2010__44_5_997_0,
author = {Bossy, Mireille and Champagnat, Nicolas and Maire, Sylvain and Talay, Denis},
title = {Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {44},
year = {2010},
pages = {997-1048},
doi = {10.1051/m2an/2010050},
mrnumber = {2731401},
zbl = {1204.82020},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2010__44_5_997_0}
}
Bossy, Mireille; Champagnat, Nicolas; Maire, Sylvain; Talay, Denis. Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 44 (2010) pp. 997-1048. doi : 10.1051/m2an/2010050. http://gdmltest.u-ga.fr/item/M2AN_2010__44_5_997_0/
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