Gradient Methods in Inverse Acoustic and Electromagnetic Scattering

The problem of determining the complex permittivity or sound speed in a bounded inhomogeneity imbedded in a homogeneous medium from scattered field measurements exterior to the inhomogeneity is considered. A number of methods of attacking this problem, all based on minimizing the difference between an integral representation of the scattered field and the measured data, are described. These include Born, Newton-Kantorovich and distorted Born methods. The main part of the paper will be devoted to a description of a gradient type algorithm which is used to minimize a cost functional in which two objective functions are sought simultaneously. The error which is minimized is a bilinear form involving the product of two functions. This special form of the nonlinearity is retained in the algorithm. A number of numerical results will be presented which illustrate the effectiveness and limitations of the approach.