  Inverse scattering has a broad applicability in quantum mechanics, remote
sensing, geophysical, and medical imaging. This paper presents a robust direct
reduced order model (ROM) method for solving inverse scattering problems based
on an efficient approximation of the resolvent operator regularizing the
Lippmann-Schwinger-Lanczos (LSL) algorithm. We show that the efficiency of the
method relies upon the weak dependence of the orthogonalized basis on the
unknown potential in the Schr\"odinger equation by demonstrating that the
Lanczos orthogonalization is equivalent to performing Gram-Schmidt on the ROM
time snapshots. We then develop the LSL algorithm in the frequency domain with
two levels of regularization. We show that the same procedure can be extended
beyond the Schr\"odinger formulation to the Helmholtz equation, e.g., to
imaging the conductivity using diffusive electromagnetic fields in conductive
media with localized positive conductivity perturbations. Numerical experiments
for Helmholtz and Schr\"odinger problems show that the proposed bi-level
regularization scheme significantly improves the performance of the LSL
algorithm, allowing for good reconstructions with noisy data and large data
sets.
