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

Mireille Bossy 1 Nicolas Champagnat 1 Sylvain Maire 2 Denis Talay 1
1 TOSCA
INRIA Lorraine, CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, INPL - Institut National Polytechnique de Lorraine, CNRS - Centre National de la Recherche Scientifique : UMR7502
2 MNC - Modélisation Numérique et Couplages
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 operator Poisson-Boltzmann equation Feynman-Kac formula random walk on sphere algorithm
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Mireille Bossy, Nicolas Champagnat, Sylvain Maire, Denis Talay. Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2010, 44 (5), pp.997-1048. ⟨inria-00459411⟩

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