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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 },
pages = {997--1048},
publisher = {EDP-Sciences},
volume = {44},
number = {5},
year = {2010},
doi = {10.1051/m2an/2010050},
mrnumber = {2731401},
zbl = {1204.82020},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2010050/}
}
TY - JOUR AU - Bossy, Mireille AU - Champagnat, Nicolas AU - Maire, Sylvain AU - Talay, Denis TI - Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2010 SP - 997 EP - 1048 VL - 44 IS - 5 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2010050/ DO - 10.1051/m2an/2010050 LA - en ID - M2AN_2010__44_5_997_0 ER -
%0 Journal Article %A Bossy, Mireille %A Champagnat, Nicolas %A Maire, Sylvain %A Talay, Denis %T Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2010 %P 997-1048 %V 44 %N 5 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2010050/ %R 10.1051/m2an/2010050 %G en %F 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 , Special Issue on Probabilistic methods and their applications, Tome 44 (2010) no. 5, pp. 997-1048. doi: 10.1051/m2an/2010050
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