Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions

Continuing our previous work [6, Inverse Problems, 2012, 28, 055002] and [5, Inverse Problems, 2012, 28, 055001],we address the ill-posedness of the inverse scattering problem ofelectromagnetic waves due to an inhomogeneous medium by studying theHessian of the data misfit. We derive and analyze the Hessian in bothHölder and Sobolev spaces. Using an integral equation approach basedon Newton potential theory and compact embeddings in Hölder andSobolev spaces, we show that the Hessian can be decomposed into threecomponents, all of which are shown to be compact operators. Theimplication of the compactness of the Hessian is that for small datanoise and model error, the discrete Hessian can be approximated by alow-rank matrix. This in turn enables fast solution of anappropriately regularized inverse problem, as well as Gaussian-basedquantification of uncertainty in the estimated inhomogeneity.