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.

翻译：逆散射在量子力学、遥感、地球物理和医学成像等领域具有广泛的适用性。本文提出了一种稳健的直接降阶模型(ROM)方法，用于求解逆散射问题，该方法基于对正则化Lippmann-Schwinger-Lanczos(LSL)算法中预解算子的高效近似。我们通过证明Lanczos正交化等价于对ROM时间快照执行Gram-Schmidt过程，表明该方法的效率依赖于正交化基对薛定谔方程中未知势函数的弱依赖性。随后，我们在频域中开发了具有两层正则化水平的LSL算法。我们证明了相同的步骤可以超越薛定谔方程体系，扩展到亥姆霍兹方程，例如利用局部正电导率扰动导电介质中的扩散电磁场对电导率进行成像。针对亥姆霍兹问题和薛定谔问题的数值实验表明，所提出的双层正则化方案显著提升了LSL算法的性能，使其能够在含噪声数据和大型数据集下实现良好的重建结果。