Optimization methods in direct and inverse scattering

In many Direct and Inverse Scattering problems one has to use a parameter-fitting procedure, because analytical inversion procedures are often not available. In this paper a variety of such methods is presented with a discussion of theoretical and computational issues. The problem of finding small subsurface inclusions from surface scattering data is solved by the Hybrid Stochastic-Deterministic minimization algorithm. A similar approach is used to determine layers in a particle from the scattering data. The Inverse potential scattering problem for spherically symmetric potentials and fixed-energy phase shifts as the scattering data is described. It is solved by the Stability Index Method. This general approach estimates the size of the minimizing sets, and gives a practically useful stopping criterion for global minimization algorithms. The 3D inverse scattering problem with fixed-energy data and its solution by the Ramm's method are discussed. An Obstacle Direct Scattering problem is treated by a Modified Rayleigh Conjecture (MRC) method. A special minimization procedure allows one to inexpensively compute scattered fields for 2D and 3D obstacles having smooth as well as nonsmooth surfaces. A new Support Function Method (SFM) is used for Inverse Obstacle Scattering problems. Another method for Inverse scattering problems, the Linear Sampling Method (LSM), is analyzed.