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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
TOSCA project-team, INRIA Sophia Antipolis – Méditerranée, France.;;
Nicolas Champagnat
TOSCA project-team, INRIA Sophia Antipolis – Méditerranée, France.;;
Sylvain Maire
IMATH, Université du sud Toulon-Var, France.
Denis Talay
TOSCA project-team, INRIA Sophia Antipolis – Méditerranée, France.;;
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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 $\mathbb{R}^d$. 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.

Research Article
© EDP Sciences, SMAI, 2010

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