Stability of solutions to inverse scattering problems with fixed-energy data

TL;DR

This paper reviews the author's results on the stability and inversion formulas for 3D inverse scattering problems with fixed-energy data, covering potential, obstacle, geophysical, and parabolic cases, including error estimates and non-uniqueness examples.

Contribution

It provides new inversion formulas, stability estimates, and an algorithm for computing the Dirichlet-to-Neumann map, advancing understanding of inverse scattering with fixed-energy data.

Findings

Derivation of inversion formulas and stability estimates for potential scattering.

Development of an algorithm for Dirichlet-to-Neumann map computation.

Construction of an example demonstrating non-uniqueness in geophysical inverse problems.

Abstract

A review of the author's results is given. Inversion formulas and stability estimates for the solutions to 3D inverse scattering problems with fixed-energy data are obtained. Inversions of exact and noisy data are stidied. The inverse potential scattering problem is discussed in detail, inversion formulas are derived and error estimates are obtained. Inverse obstacle scattering problem with data at a fixed frequency is studied. Uniqueness theorems and stability estimates are obtained. Inverse geophysical scattering problem is discussed. An algorithm for computing the Dirichlet-to-Neumann map from the scattering amplitude and vicxe versa is obtained. An analytical example of non-uniqueness of the solution to a 3D inverse geophysical problem is constructed. An inverse problem for a parabolic equation is discussed.