Numerical solution of an inverse medium scattering problem with a stochastic source

This paper is concerned with the inverse medium scattering problem with a stochastic source, the reconstruction of the refractive index of an inhomogeneous medium from the boundary measurements of the scattered field. As an inverse problem, there are two major difficulties in addition to being highly nonlinear: the ill-posedness and the presence of many local minima. To overcome the difficulties, a stable and efficient recursive linearization method has been recently developed for solving the inverse medium scattering problem with a deterministic source. Compared to the classical inverse problems, stochastic inverse problems, referred to the inverse problems involving uncertainties, have substantially more difficulties due to the randomness and uncertainties. Based on Wiener chaos expansion, the stochastic problem is converted into a set of decoupled deterministic problems. The strategy is developed a new hybrid method combining the Wiener chaos expansion with the recursive linearization method for solving the inverse medium problem with a stochastic source. Numerical experiments are reported to demonstrate the effectiveness of the proposed approach.