In this paper we develop a numerical method for solving an inverse scattering problem of estimating the scattering potential in a Schr\"{o}dinger equation from frequency domain measurements based on reduced order models (ROM). The ROM is a projection of Schr\"{o}dinger 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 non-linear 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.

翻译：本文提出了一种数值方法，用于解决从频域测量数据中估计薛定谔方程散射势的逆散射问题，该方法基于降阶模型（ROM）。该ROM是将薛定谔算子投影到由其特定波数下的解快照所张成的子空间上。只要测量在这些波数下进行，ROM就可以根据散射体周围表面上的测量数据以数据驱动的方式构建。一旦计算出ROM，就可以通过最小化ROM失配的非线性优化来估计散射势。这种方法通常优于基于数据失配最小化的传统方法。我们开发了两种基于ROM的逆散射算法变体，并在二维空间的一个合成算例上进行了测试。