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Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics

Published online by Cambridge University Press:  26 August 2010

Mireille Bossy ,
Nicolas Champagnat ,
Sylvain Maire  and
Denis Talay
Mireille Bossy
Affiliation:
TOSCA project-team, INRIA Sophia Antipolis – Méditerranée, France. Mireille.Bossy@sophia.inria.fr; Nicolas.Champagnat@sophia.inria.fr; Denis.Talay@sophia.inria.fr
Nicolas Champagnat
Affiliation:
TOSCA project-team, INRIA Sophia Antipolis – Méditerranée, France. Mireille.Bossy@sophia.inria.fr; Nicolas.Champagnat@sophia.inria.fr; Denis.Talay@sophia.inria.fr
Sylvain Maire
Affiliation:
IMATH, Université du sud Toulon-Var, France. maire@univ-tln.fr
Denis Talay
Affiliation:
TOSCA project-team, INRIA Sophia Antipolis – Méditerranée, France. Mireille.Bossy@sophia.inria.fr; Nicolas.Champagnat@sophia.inria.fr; Denis.Talay@sophia.inria.fr
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Abstract

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 $\mathbb{R}^d$. 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.

Keywords

Divergence form operatorPoisson-Boltzmann equationFeynman-Kac formularandom walk on sphere algorithm
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 5: Special Issue on Probabilistic methods and their applications , September 2010 , pp. 997 - 1048
Copyright
© EDP Sciences, SMAI, 2010

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