Spectral approximation to a transmission eigenvalue problem and its applications to an inverse problem

We first develop an efficient spectral-Galerkin method and a rigorous error analysis for the generalized eigenvalue problems associated to a transmission eigenvalue problem. Then, we present an iterative scheme, based on computation of the first transmission eigenvalue, to estimate the index of refraction of an inhomogeneous medium. We present ample numerical results to demonstrate the effectiveness and accuracy of our approach.