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 these 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 classical inverse problems, stochastic inverse problems, referred to as inverse problems involving uncertainties, have substantially more difficulties due to randomness and uncertainties. Based on the Wiener chaos expansion, the stochastic problem is converted into a set of decoupled deterministic problems. The strategy developed is a new hybrid method combining the WCE 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.