In this paper we develop a numerical method for solving an inverse scattering problem of estimating the scattering potential in a Schrödinger equation from frequency domain measurements based on reduced order models (ROM). The ROM is a projection of the Schrödinger operator onto a subspace spanned by its solution snapshots at certain wavenumbers. Provided the measurements are performed at these wavenumbers, the ROM can be constructed in a data-driven manner from the measurements on a surface surrounding the scatterers. Once the ROM is computed, the scattering potential can be estimated using nonlinear optimization that minimizes the ROM misfit. Such an approach typically outperforms the conventional methods based on data misfit minimization. We develop two variants of ROM-based algorithms for inverse scattering and test them on a synthetic example in two spatial dimensions.

We study a reduced order model (ROM) based waveform inversion method applied to a Helmholtz problem with impedance boundary conditions and variable refractive index. The first goal of this paper is to obtain relations that allow the reconstruction of the Galerkin projection of the continuous problem onto the space spanned by solutions of the Helmholtz equation. The second goal is to study the introduced nonlinear optimization method based on the ROM aimed to estimate the refractive index from reflection and transmission data. Finally we compare numerically our method to the conventional least squares inversion based on minimizing the distance between modelled to measured data.