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Probabilistic interpretation and random walkon spheres algorithms for the Poisson-Boltzmann equationin 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 methodsfor PDE models in molecular dynamics,we establish a new probabilistic interpretation of a family of divergence formoperators with discontinuous coefficients at the interfaceof two open subsets of $\mathbb{R}^d$ . This family of operators includes the case of thelinearized Poisson-Boltzmann equation used tocompute the electrostatic free energy of a molecule.More precisely, we explicitly construct a Markov process whoseinfinitesimal generator belongs to this family, as the solution of a SDEincluding a non standard local time term related to the interfaceof discontinuity. We then prove an extendedFeynman-Kac formula for the Poisson-Boltzmann equation.This formula allows us to justifyvarious probabilistic numerical methods toapproximate the free energy of a molecule.We analyse the convergence rate of these simulation procedures andnumerically compare them on idealized molecules models.

Keywords

Divergence form operatorPoisson-Boltzmann equationFeynman-Kac formularandom walk on sphere algorithm

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