  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.
