We present a reduced-order model (ROM) methodology for inverse scattering problems in which the reduced-order models are data-driven, i.e. they are constructed directly from data gathered by sensors. Moreover, the entries of the ROM contain localised information about the coefficients of the wave equation. We solve the inverse problem by embedding the ROM in physical space. Such an approach is also followed in the theory of ``optimal grids,'' where the ROMs are interpreted as two-point finite-difference discretisations of an underlying set of equations of a first-order continuous system on this special grid. Here, we extend this line of work to wave equations and introduce a new embedding technique, which we call Krein embedding, since it is inspired by Krein's seminal work on vibrations of a string. In this embedding approach, an adaptive grid and a set of medium parameters can be directly extracted from a ROM and we show that several limitations of optimal grid embeddings can be avoided. Furthermore, we show how Krein embedding is connected to classical optimal grid embedding and that convergence results for optimal grids can be extended to this novel embedding approach. Finally, we also briefly discuss Krein embedding for open domains, that is, semi-infinite domains that extend to infinity in one direction.

翻译：我们提出了一种针对逆散射问题的降阶模型（ROM）方法，其中降阶模型是数据驱动的，即直接由传感器采集的数据构建。此外，ROM的条目包含波动方程系数的局部化信息。我们通过将ROM嵌入物理空间来求解逆问题。这种方法也遵循“最优网格”理论的思路，在该理论中，ROM被解释为底层一阶连续系统方程组在该特殊网格上的两点有限差分离散化。在此，我们将这一工作路线扩展到波动方程，并引入一种新的嵌入技术，称之为Krein嵌入，因其灵感来自Krein关于弦振动的开创性工作。在这种嵌入方法中，自适应网格和一组介质参数可以直接从ROM中提取，并且我们表明可以避免最优网格嵌入的若干局限性。此外，我们展示了Krein嵌入如何与经典的最优网格嵌入相关联，并且最优网格的收敛性结果可以推广到这种新的嵌入方法。最后，我们还简要讨论了开放域（即沿一个方向延伸至无穷远的半无限域）上的Krein嵌入。