A numerical study of single source localization algorithms for phaseless inverse scattering problems

Phaseless inverse scattering problems appear often in practical applications since phaseless data are relatively easier to measure than the phased data, but they are also numerically more difficult to solve due to the translation invariance property. Based on three distinct noisy measurements of phaseless far-field data, the phase information can be approximately reconstructed by formulating it as a single source localization problem, for which many efficient algorithms are readily available. In this paper, we numerically compare several source localization algorithms based on different norm formulations in the context of inverse scattering. As one major contribution, we propose an improved phase retrieval algorithm, which addresses some pitfalls of the original phase retrieval algorithm in [X. Ji, X. Liu, B. Zhang, SIAM J. Imaging Sci. 12 (1) (2019) 372–391.] Moreover, a simple criterion of minimizing the condition number of the underlying linear least square system is advocated for optimizing the choices of scattering strengths (or sensors’ locations). Extensive numerical results are shown to illustrate the similarity and difference among the tested algorithms.