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

Abstract

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 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.Mathematics Subject Classification. 35Q60, 92C40, 60J60, 65C05, 65C20, 68U20.

Simulating diffusion processes in discontinuous media: A numerical scheme with constant time steps

“…Discontinuities of D happen in many scientific domains, where the question of the simulation of X has been treated: geophysics [1][2][3]43,11,17,27,32,33,31,41,[54][55][56]59,62,5], ecology [13,53,57], brain imaging [22], astrophysics [42,64], meteorology [61], molecular dynamic [47,9], . .…”

Simulating diffusions with piecewise constant coefficients using a kinetic approximation

“…With Remark 2, it can provide the law of the exit time from the domain, which is crucial in the computation of the principal eigenelements of the operator by means of Monte Carlo methods [28,29]. In a recent work [5] it has also been used successfully for solving the Poisson-Boltzmann equation of molecular electrostatics for which the finite differences method was originally designed. It has been proved for example that in the case of a molecule constituted of one sphere our method is of order two in ε instead of order one for the finite differences method.…”

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