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We analyze the inverse problem to reconstruct the shape of a threedimensional homogeneous dielectric obstacle from the knowledge of noisy farfield data. The forward problem is solved by a system of second kind boundaryintegral equations. For the numerical solution of these coupled integralequations we propose a fast spectral algorithm by transporting these equationsonto the unit sphere. We review the differentiability properties of theboundary to far field operator and give a characterization of the adjointoperator of the first Fr\'echet derivative. Using these results we discuss theimplementation of the iteratively regularized Gauss-Newton method for thenumerical solution of the inverse problem and give numerical results forstar-shaped obstacles.

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