4 research outputs found
Stochastic finite differences for elliptic diffusion equations in stratified domains
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
A Partially Reflecting Random Walk on Spheres Algorithm for Electrical Impedance Tomography
In this work, we develop a probabilistic estimator for the voltage-to-current
map arising in electrical impedance tomography. This novel so-called partially
reflecting random walk on spheres estimator enables Monte Carlo methods to
compute the voltage-to-current map in an embarrassingly parallel manner, which
is an important issue with regard to the corresponding inverse problem. Our
method uses the well-known random walk on spheres algorithm inside subdomains
where the diffusion coefficient is constant and employs replacement techniques
motivated by finite difference discretization to deal with both mixed boundary
conditions and interface transmission conditions. We analyze the global bias
and the variance of the new estimator both theoretically and experimentally. In
a second step, the variance is considerably reduced via a novel control variate
conditional sampling technique
Monte Carlo methods for linear and non-linear Poisson-Boltzmann equation
The electrostatic potential in the neighborhood of a biomolecule can be
computed thanks to the non-linear divergence-form elliptic Poisson-Boltzmann
PDE. Dedicated Monte-Carlo methods have been developed to solve its linearized
version (see e.g.Bossy et al 2009, Mascagni & Simonov 2004}). These algorithms
combine walk on spheres techniques and appropriate replacements at the boundary
of the molecule. In the first part of this article we compare recent
replacement methods for this linearized equation on real size biomolecules,
that also require efficient computational geometry algorithms. We compare our
results with the deterministic solver APBS. In the second part, we prove a new
probabilistic interpretation of the nonlinear Poisson-Boltzmann PDE. A Monte
Carlo algorithm is also derived and tested on a simple test case