236 research outputs found
New Monte Carlo schemes for simulating diffusions in discontinuous media
International audienceWe introduce new Monte Carlo simulation schemes for diffusions in a discontinuous media divided in subdomains with piecewise constant diffusivity. These schemes are higher order extensions of the usual schemes and take into account the two dimensional aspects of the diffusion at the interface between subdomains. This is achieved using either stochastic processes techniques or an approach based on finite differences. Numerical tests on elliptic, parabolic and eigenvalue problems involving an operator in divergence form show the efficiency of these new schemes
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
Importance Sampling for Multiscale Diffusions
We construct importance sampling schemes for stochastic differential
equations with small noise and fast oscillating coefficients. Standard Monte
Carlo methods perform poorly for these problems in the small noise limit. With
multiscale processes there are additional complications, and indeed the
straightforward adaptation of methods for standard small noise diffusions will
not produce efficient schemes. Using the subsolution approach we construct
schemes and identify conditions under which the schemes will be asymptotically
optimal. Examples and simulation results are provided
Simulating diffusions with piecewise constant coefficients using a kinetic approximation
International audienceUsing a kinetic approximation of a linear diffusion operator, we propose an algorithm that allows one to deal with the simulation of a multi-dimensional stochastic process in a media which is locally isotropic except on some surface where the diffusion coefficient presents some discontinuities. Numerical examples are given in dimensions one to three on PDEs or stochastic PDEs with or without source terms
Interfacial Phenomena and Natural Local Time
This article addresses a modification of local time for stochastic processes,
to be referred to as `natural local time'. It is prompted by theoretical
developments arising in mathematical treatments of recent experiments and
observations of phenomena in the geophysical and biological sciences pertaining
to dispersion in the presence of an interface of discontinuity in dispersion
coefficients. The results illustrate new ways in which to use the theory of
stochastic processes to infer macro scale parameters and behavior from micro
scale observations in particular heterogeneous environments
On the constructions of the skew Brownian motion
This article summarizes the various ways one may use to construct the Skew
Brownian motion, and shows their connections. Recent applications of this
process in modelling and numerical simulation motivates this survey. This
article ends with a brief account of related results, extensions and
applications of the Skew Brownian motion.Comment: Published at http://dx.doi.org/10.1214/154957807000000013 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
- …