In this paper, we extend the ROM-based approach for inverse scattering with Neumann boundary conditions, introduced by Druskin at. al. (Inverse Problems 37, 2021), to the 1D Schr{\"o}dinger equation with impedance (Robin) boundary conditions. We also propose a novel data-assimilation (DA) inversion method based on the ROM approach, thereby avoiding the need for a Lanczos-orthogonalization (LO) step. Furthermore, we present a detailed numerical study and comparison of the accuracy and stability of the DA and LO methods.

翻译：本文我们将 Druskin 等人（《逆问题》第37卷，2021年）提出的基于 Neumann 边界条件的 ROM 逆散射方法扩展至带阻抗（Robin）边界条件的一维薛定谔方程。同时，提出一种基于 ROM 方法的新型数据同化（DA）反演方法，从而避免兰佐斯正交化（LO）步骤。此外，我们通过详细的数值研究，对比分析了 DA 与 LO 方法的精度与稳定性。