Inverse scattering via skin effect

We present a stable method for the inverse scattering problem of the Helmholtz equation in two dimensions. The algorithm requires single-frequency scattering data, and is an iterative procedure resembling the process of layer stripping. The method is based on the observation that the ill-posedness of the inverse scattering problem causes it to be almost linear in certain regimes. There the algorithm solves the linearized equations to produce an approximate scatterer within a narrow layer beneath the surface of the scatterer. This approximation is used to linearize equations governing an inner narrow layer beneath the first layer. While the linearized equations are solved and an approximate scatterer produced in the second layer, the previously obtained approximation in the first layer is refined. The procedure is repeated until the entire scatterer is approximated and the approximation refined. Several numerical examples are presented to demonstrate the performance of the algorithm in the special case of radially symmetric scatterers.