Volume 44, Number 5, September-October 2010Special Issue on Probabilistic methods and their applications
|Page(s)||997 - 1048|
|Published online||26 August 2010|
Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics
TOSCA project-team, INRIA Sophia Antipolis –
Méditerranée, France. Mireille.Bossy@sophia.inria.fr; Nicolas.Champagnat@sophia.inria.fr; Denis.Talay@sophia.inria.fr
2 IMATH, Université du sud Toulon-Var, France. firstname.lastname@example.org
Revised: 23 February 2010
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.
Mathematics Subject Classification: 35Q60 / 92C40 / 60J60 / 65C05 / 65C20 / 68U20
Key words: Divergence form operator / Poisson-Boltzmann equation / Feynman-Kac formula / random walk on sphere algorithm
© EDP Sciences, SMAI, 2010
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