An iterative approach to monochromatic phaseless inverse scattering
This work addresses the challenging inverse problem of phase retrieval in scattering for potential reconstruction, offering a computationally efficient iterative scheme with theoretical convergence guarantees at high energies.
The paper proposes an iterative method for recovering a Schrödinger potential from phaseless scattering data using additional measurements with known background objects. The method converges globally at high energies and yields accurate approximations at moderate energies with low computational cost.
This paper is concerned with the inverse problem to recover a compactly supported Schr{ö}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.