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

Mireille Bossy; Nicolas Champagnat; Sylvain Maire; Denis Talay

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 5, page 997-1048
  • ISSN: 0764-583X

Abstract

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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 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.

How to cite

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Bossy, Mireille, et al. "Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics." ESAIM: Mathematical Modelling and Numerical Analysis 44.5 (2010): 997-1048. <http://eudml.org/doc/250790>.

@article{Bossy2010,
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. },
author = {Bossy, Mireille, Champagnat, Nicolas, Maire, Sylvain, Talay, Denis},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Divergence form operator; Poisson-Boltzmann equation; Feynman-Kac formula; random walk on sphere algorithm; divergence form operator; Feyman-Kac formula; random walk on sphere},
language = {eng},
month = {8},
number = {5},
pages = {997-1048},
publisher = {EDP Sciences},
title = {Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics},
url = {http://eudml.org/doc/250790},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Bossy, Mireille
AU - Champagnat, Nicolas
AU - Maire, Sylvain
AU - Talay, Denis
TI - Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/8//
PB - EDP Sciences
VL - 44
IS - 5
SP - 997
EP - 1048
AB - 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.
LA - eng
KW - Divergence form operator; Poisson-Boltzmann equation; Feynman-Kac formula; random walk on sphere algorithm; divergence form operator; Feyman-Kac formula; random walk on sphere
UR - http://eudml.org/doc/250790
ER -

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