An iterative approach to monochromatic phaseless inverse scattering

TL;DR

This paper introduces an iterative method for reconstructing Schr{"o}dinger potentials from phaseless scattering data, utilizing additional measurements with known backgrounds, and demonstrates high-order convergence at high energies.

Contribution

It proposes a novel iterative scheme for phaseless inverse scattering that converges globally at high energies and improves reconstructions with regularization at moderate energies.

Findings

The method converges globally at high energies depending on potential smoothness.

Numerical experiments show high accuracy with low computational effort.

Small noise levels can be mitigated with additional regularization steps.

Abstract

This paper is concerned with the inverse problem to recover a compactly supported Schr{\"o}dinger potential given the differential scattering cross section, i.e. the modulus, but not the phase of the scattering amplitude. To compensate for the missing phase information we assume additional measurements of the differential cross section in the presence of known background objects. We propose an iterative scheme for the numerical solution of this problem and prove that it converges globally of arbitrarily high order depending on the smoothness of the unknown potential as the energy tends to infinity. At fixed energy, however, the proposed iteration does not converge to the true solution even for exact data. Nevertheless, numerical experiments show that it yields remarkably accurate approximations with small computational effort even for moderate energies. At small noise levels it may be worth to improve these approximations by a few steps of a locally convergent iterative regularization method, and we demonstrate to which extent this reduces the reconstruction error.