Nonlinear approach to inverse scattering

The inverse scattering problem for the scalar wave equation associated with propagation through a medium whose index of refraction differs from that of free space in a region of compact support is treated when the scattered data is given for diverse directions of (plane wave) incidence, scattered directions, and frequencies. The problem is formulated in terms of the minimization of a nonlinear functional which is bounded below, subject to constraints. It is shown that the conditional-gradient method may be employed, the iteration process converging to stationary points. The linearized version (corresponding to the perturbed wave equation with only the linear perturbed terms retained) of the nonlinear functional is considered as a special case. In particular the linearized version related to the Born approximation leads to some additional new results.