A spectrally accurate method for the dielectric obstacle scattering problem and applications to the inverse problem

We analyze the inverse problem to reconstruct the shape of a three dimensional homogeneous dielectric obstacle from the knowledge of noisy far field data. The forward problem is solved by a system of second kind boundary integral equations. For the numerical solution of these coupled integral equations we propose a fast spectral algorithm by transporting these equations onto the unit sphere. We review the differentiability properties of the boundary to far field operator and give a characterization of the adjoint operator of the first Fr\'echet derivative. Using these results we discuss the implementation of the iteratively regularized Gauss-Newton method for the numerical solution of the inverse problem and give numerical results for star-shaped obstacles.