Exploring Low-Rank Structure for an Inverse Scattering Problem with Far-Field Data

Abstract. In this work, we introduce a novel low-rank structure tailored for solving the inverse scattering problem. The particular low-rank structure is given by the generalized prolate spheroidal wave functions, computed stably and accurately via a Sturm–Liouville problem. We first process the far-field data to obtain a postprocessed data set within a disk domain. Subsequently, the postprocessed data are projected onto a low-rank space given by the low-rank structure. The unknown is approximately solved in this low-rank space by dropping higher-order terms. The low-rank structure leads to an explicit stability estimate for unknown functions belonging to standard Sobolev spaces and a Lipschitz stability estimate for unknowns belonging to a finite-dimensional low-rank space. Various numerical experiments are conducted to validate its performance, encompassing assessments of resolution capability, robustness against randomly added noise and modeling errors, and demonstration of increasing stability.