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 approximations of the Neumann problem

We introduce Monte Carlo methods to compute the solution of elliptic equations with pure Neumann boundary conditions. We first prove that the solution obtained by the stochastic representation has a zero mean value with respect to the invariant measure of the stochastic process associated to the equation. Pointwise approximations are computed by means of standard and new simulation schemes especially devised for local time approximation on the boundary of the domain. Global approximations are computed thanks to a stochastic spectral formulation taking into account the property of zero mean value of the solution. This stochastic formulation is asymptotically perfect in terms of conditioning. Numerical examples are given on the Laplace operator on a square domain with both pure Neumann and mixed Dirichlet-Neumann boundary conditions. A more general convection-diffusion equation is also numerically studied

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

Reply to the correspondence: "On the fracture toughness of bioinspired ceramic materials"

This is a reply to the correspondence of Prof. Robert Ritchie: "On the fracture toughness of bioinspired ceramic materials", submitted to Nature Materials, which discusses the fracture toughness values of the following papers: Bouville, F., Maire, E., Meille, S., Van de Moort\`ele, B., Stevenson, A. J., & Deville, S. (2014). Strong, tough and stiff bioinspired ceramics from brittle constituents. Nature Materials, 13(5), 508-514 and Le Ferrand, H., Bouville, F., Niebel, T. P., & Studart, A. R. (2015). Magnetically assisted slip casting of bioinspired heterogeneous composites. Nature Materials, 14(11), 1172-1172.Comment: 5 pages, 2 figure

The walk on moving spheres: a new tool for simulating Brownian motion's exit time from a domain

International audienceIn this paper we introduce a new method for the simulation of the exit time and position of a $\delta$-dimensional Brownian motion from a domain. The main interest of our method is that it avoids splitting time schemes as well as inversion of complicated series. The idea is to use the connexion between the $\delta$-dimensional Bessel process and the $\delta$-dimensional Brownian motion thanks to an explicit Bessel hitting time distribution associated with a particular curved boundary. This allows to build a fast and accurate numerical scheme for approximating the hitting time. Numerical comparisons with existing methods are performed

A Study of Biased and Unbiased Stochastic Algorithms for Solving Integral Equations

In this paper we propose and analyse a new unbiased stochastic method for solving a class of integral equations, namely the second kind Fredholm integral equations. We study and compare three possible approaches to compute linear functionals of the integral under consideration: i) biased Monte Carlo method based on evaluation of truncated Liouville-Neumann series, ii) transformation of this problem into the problem of computing a finite number of integrals, and iii) unbiased stochastic approach. Five Monte Carlo algorithms for numerical integration have been applied for approach (ii). Error balancing of both stochastic and systematic errors has been discussed and applied during the numerical implementation of the biased algorithms. Extensive numerical experiments have been performed to support the theoretical studies regarding the convergence rate of Monte Carlo methods for numerical integration done in our previous studies. We compare the results obtained by some of the best biased stochastic approaches with the results obtained by the proposed unbiased approach. Conclusions about the applicability and efficiency of the algorithms have been drawn

A New Walk on Equations Monte Carlo Method for Linear Algebraic Problems

International audienceA new Walk on Equations (WE) Monte Carlo algorithm for Linear Algebra (LA) problem is proposed and studied. This algorithm relies on a non-discounted sum of an absorbed random walk. It can be applied for either real or complex matrices. Several techniques like simultaneous scoring or the sequential Monte Carlo method are applied to improve the basic algorithm. Numerical tests are performed on examples with matrices of different size and on systems coming from various applications. Comparisons with standard deterministic or Monte Carlo algorithms are also done