Stability of the solutions to 3D inverse scattering problems with fixed-energy data.

Abstract

A review of the author’s results is given. Inversion formulas and stability results for the solutions to 3D inverse scattering problems with fixed energy data are obtained. Inversion of exact and noisy data is considered. The inverse potential scattering problem with fixed-energy scattering data is discussed in detail, inversion formulas for the exact and for noisy data are derived, error estimates for the inversion formulas are obtained. The inverse obstacle scattering problem is considered for non-smooth obstacles. Stability estimates are derived for inverse obstacle scattering problem in the class of smooth obstacles. Global estimates for the scttering amplitude are given when the potential grows to infinity in a bounded domain. Inverse geophysical scattering problem is discussed briefly. An algorithm for constructing the Dirichlet-to-Neumann map from the scattering amplitude and vice versa is obtained. An analytical example of non-uniqueness of the solution to a 3D inverse problem of geophysics and a uniqueness theorem for an inverse problem for parabolic equations are given.