Reconstruction of a penetrable obstacle in periodic waveguides

The reconstruction of a penetrable obstacle embedded in a periodic waveguide is a challenging problem. In this paper, the inverse problem is formulated as an optimization problem. We prove some properties of the scattering operator and propose an iterative scheme to approximate the support of the obstacle. Using the limiting absorption principle and a recursive doubling technique, we implement a fast algorithm based on a carefully designed finite element method for the forward scattering problem. Numerical examples validate the effectiveness of the method.