An efficient method for the solution of the inverse scattering problem for penetrable obstacles.

We consider a two-dimensional scattering problem for inhomogeneous media. This problem arises from the study of timeharmonic electromagnetic scattering problems for Transverse Magnetic (TM) waves. In the direct scattering problem we have to compute the scattered wave from the knowledge of the incident wave and of the inhomogeneity. In the inverse scattering problem we have to reconstruct the refractive index of the inhomogeneity from some knowledge of the scattered waves generated by the inhomogeneity itself with known incident waves. The complete formulation of this inverse problem is given by a system of two integral equations, where the unknown functions are the refractive index of the inhomogeneity, and the total field inside the inhomogeneity. Note that, the total field is defined as the sum of the incident field plus the scattered field. The numerical solution of this system has a high computational cost even for low or medium size discretization schemes. The main contribution of the present paper is given by an efficient method for the numerical solution of this problem, where, roughly speaking, the computation of the total field inside the inhomogeneity is avoided. Some numerical experiments are used to show the performance of the proposed method. In these experiments, we have used exact scattering data, and synthetic scattering data.