Fast numerical solution of the electromagnetic medium scattering problem and applications to the inverse problem

We propose a method for computing the scattering of a time-harmonic electromagnetic wave in a medium with a locally perturbed refractive index. We prove that it converges of arbitrarily high order depending on the smoothness of the refractive index and that the time complexity is O(N3 log(N)). As an application we solve the corresponding inverse problem using a preconditioned Newton method and discuss how to compute the Fréchet derivative of the solution operator and its adjoint by the proposed method. The performance of both the forward and the inverse solver is illustrated in a number of numerical experiments.