Monte Carlo methods for discontinuous scattering: application to electrical impedance tomography.

Summary This thesis deals with the development of Monte Carlo methods for computing Feynman-Kac representations involving operators in divergence form with a piecewise constant diffusion coefficient. The proposed methods are variants of the walk on spheres within zones with a constant diffusion coefficient and stochastic finite difference techniques to deal with interface conditions as well as boundary conditions of different types. By combining these two techniques, we obtain random walks whose score computed along the path provides a biased estimator of the solution of the partial differential equation considered. We show that the global bias of our algorithm is in general of order two with respect to the finite difference step. These methods are then applied to the direct problem related to electrical impedance tomography for tumor detection. A variance reduction technique is also proposed in this framework. Finally, the inverse problem of tumor detection from surface measurements is addressed using two stochastic algorithms based on a parametric representation of the tumor or tumors as one or more spheres. Numerous numerical tests are proposed and show convincing results in the localization of tumors.