A data-driven approach to solving a 1D inverse scattering problem

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.