Numerical approximation of the potential in the two-dimensional inverse scattering problem

TL;DR

This paper introduces an iterative numerical method for approximating the potential in a 2D inverse scattering problem using various scattering data types, extending previous fixed energy algorithms with new numerical results.

Contribution

It develops a new iterative algorithm for the 2D inverse scattering problem that handles multiple data types, building on and extending Novikov's fixed energy approach.

Findings

Numerical results align with theoretical estimates for large energy data.

The algorithm effectively reconstructs potentials from different scattering data types.

The method demonstrates robustness across various scattering scenarios.

Abstract

We present an iterative algorithm to compute numerical approximations of the potential for the Schr\"odinger operator from scattering data. Four different types of scattering data are used as follows: fixed energy, fixed incident angle, backscattering and full data. In the case of fixed energy, the algorithm coincides basically with the one recently introduced by Novikov in [Novikov, R. G., "An iterative approach to non-overdetermined inverse scattering at fixed energy", Sbornik: Mathematics 206 (1), 120-134 (2015)], where some estimates are obtained for large energy scattering data. The numerical results that we present here are consistent with these estimates.