Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics
ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 5, pp. 997-1048.

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 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.

DOI : 10.1051/m2an/2010050
Classification : 35Q60, 92C40, 60J60, 65C05, 65C20, 68U20
Mots clés : divergence form operator, Poisson-Boltzmann equation, Feynman-Kac formula, random walk on sphere algorithm
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     title = {Probabilistic interpretation and random walk on spheres algorithms for the {Poisson-Boltzmann} equation in molecular dynamics},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
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Bossy, Mireille; Champagnat, Nicolas; Maire, Sylvain; Talay, Denis. Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics. ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 5, pp. 997-1048. doi : 10.1051/m2an/2010050. http://www.numdam.org/articles/10.1051/m2an/2010050/

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