Stochastic finite differences for elliptic diffusion equations in stratified domains

Abstract

International audienceWe describe Monte Carlo algorithms to solve elliptic partial differen- tial equations with piecewise constant diffusion coefficients and general boundary conditions including Robin and transmission conditions as well as a damping term. The treatment of the boundary conditions is done via stochastic finite differences techniques which possess an higher order than the usual methods. The simulation of Brownian paths inside the domain relies on variations around the walk on spheres method with or without killing. We check numerically the efficiency of our algorithms on various examples of diffusion equations illustrating each of the new techniques introduced here

Monte Carlo methods for linear and non-linear Poisson-Boltzmann equation

“…Totally Asymmetric Jump (TAJ): This jump method from Γ is a new proposition in the context of the Poisson-Boltzmann equation. It was originally proposed in a two-dimensional context by Lejay and Maire [22] and for more general equations and boundary conditions by Maire and Nguyen in [24]. These methods apply to linear divergence form equations with damping.…”

“…While the proof is based on similar computations in [24], for the sake of completeness we give a detailed proof in the context of Poisson Boltzmann equation. Note that the above TAJ replacement formulas are more convenient than the formulas derived in [24], as they do not require to impose some constraints on h.…”

“…These MC algorithms involve double randomization techniques [27], walk on spheres techniques [30] to simulate Brownian motion exit times and positions in Ω out and Ω in , and asymmetric jump methods from the boundary Γ to take into account the discontinuity of ε in the divergence form of the PDE (1.5). Recently, Lejay and Maire [22] and Maire and Nguyen [24] have proposed new replacement methods from the boundary Γ which improve the order of convergence of the original algorithm. Section 2 is devoted to MC algorithms for the linearized Poisson-Boltzmann equation.…”

Anomaly Detection in Random Heterogeneous Media

“…In the last decade, the numerical simulation of diffusion processes with discontinuous coefficients has been the subject of active research, cf., e.g., [25,45,46,47,106,108,110,116,117,120,122]. While the one-dimensional situation is well understood, the question whether there exist exact simulation schemes for the multi-dimensional case is still unsettled.…”

A partially reflecting random walk on spheres algorithm for electrical impedance tomography

“…Also for the simulation of diffusion processes in discontinuous media, second order schemes were proposed and analyzed by Bossy et al [5] and by Lejay and the first author [25]. The idea to use a local finite difference discretization for the simulation of the boundary behavior of reflecting diffusion processes was introduced recently by the first author and Tanré [30] and further developed by the first author and Nguyen [31]. Other, related schemes were defined by Lejay and Pichot [26] and Lejay and the first author [28].…”