Optimization Methods in Direct and Inverse Scattering

Solving inverse problems, in particular inverse scattering problems, parameterfitting procedures are used quite often because the analytical inversion procedures are not available. The general scheme for such procedures is simple: one has a relation B(q) = A, where B is some operator, q is an unknown function, and A is the data. In inverse scattering problems q is a potential, and A is the scattering amplitude. If q is sought in a finite-parametric family of functions, then q = q(x, p), where p = (p1, ...., pn) is a parameter. The parameter is found by solving a global minimization problem: Φ[B(q(x, p))−A] = min, where Φ is some positive functional, and q ∈ Q, where Q is an admissible set of q. In practice the above problem often has many local minimizers, and the global minimizer is not ncessarily unique. Moreover, as a rule, the data A are known with some error. Thus Aδ is known, such that ||A−Aδ|| < δ. There are no stability estimates which would show how the global minimizer q(x, popt) is perturbed when the data A are replaced by the perturbed data Aδ. In fact, one can easily construct examples showing that there is no stability of the global minimizer with respect to small errors in the data, in general. By these reasons practically there is no guarantee that the parameterfitting procedures will yield a solution to the inverse problem with a guaranteed accuracy. However, overwhelming majority of practitioners are using parameterfitting procedures. In dozens of published papers the results obtained by various parameter-fitting procedures look quite good. The explanation, in most of the cases is simple: the authors know the answer beforehand, and it is usually not difficult to parametrize the unknown function so that the exact solution is well approximated by the function from finite-parametric family, and since the authors know a priori the exact answer, they may choose numerically the values of the parameters which yield a good approximation of the exact solution. When can one rely on the results obtained by parameter-fitting procedures? Unfortunately, there is no rigorous and complete answer to this question. We propose a practically useful answer, which consists of the following method, called the Stability Index method.