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

In this paper, we extend a recently proposed approach for inverse scattering with Neumann boundary conditions [Druskin et al., Inverse Probl. 37, 075003 (2021)] to the 1D Schrodinger equation with impedance (Robin) boundary conditions. This method approaches inverse scattering in two steps: first, to extract a reduced order model (ROM) directly from the data and, subsequently, to extract the scattering potential from the ROM. 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 A comparison of the accuracy and stability of the DA and LO methods.

Automated summary

This paper extends a recently proposed approach for inverse scattering to the 1D Schrodinger equation with impedance boundary conditions. The method involves extracting a reduced order model directly from the data and then using it to extract the scattering potential. A novel data-assimilation inversion method based on the reduced order model approach is also proposed, which eliminates the need for a Lanczos-orthogonalization step. Furthermore, a detailed numerical study and comparison of the accuracy and stability of the data-assimilation and Lanczos-orthogonalization methods are presented.