The inverse scattering problem exhibits an inherent low-rank structure due to its ill-posed nature; however developing low-rank structures for the inverse scattering problem remains challenging. 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 post-processed data set within a disk domain. Subsequently, the post-processed 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 a H\"{o}lder-logarithmic type stability estimate for arbitrary unknown functions, 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.

翻译：逆散射问题因其不适定性而展现出固有的低秩结构；然而，为逆散射问题构建低秩结构仍具挑战性。本文提出一种专为求解逆散射问题设计的新型低秩结构。该特定低秩结构由广义扁球面波函数给出，可通过斯图姆-刘维尔问题稳定且精确地计算。我们首先处理远场数据，在圆盘域内获得后处理数据集。随后，将后处理数据投影到由该低秩结构定义的低秩空间上。通过舍弃高阶项，未知量在此低秩空间中得到近似求解。该低秩结构可导出任意未知函数的赫尔德-对数型稳定性估计，以及对属于有限维低秩空间的未知量的利普希茨稳定性估计。通过开展多项数值实验验证其性能，包括分辨率评估、抗随机添加噪声与建模误差的鲁棒性测试，以及稳定性递增现象的展示。