Together with my colleague Professor Sylvain Maire from Université du Toulon, France, we have developed some extensions of the well-known random walk on spheres estimator to simulate reflecting and partially reflecting diffusion processes. In our recent article which has been published in the Journal of Computational Physics we adapt these techniques to the forward problem of electrical impedance tomography.
Here comes the abstract: 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. Subsequently, the variance of the new estimator is considerably reduced via a novel control variate conditional sampling technique which yields a highly efficient hybrid forward solver coupling probabilistic and deterministic algorithms.