We generate data-driven reduced order models (ROMs) for inversion of the one and two dimensional Schrodinger equation in the spectral domain given boundary data at a few frequencies. The ROM is the Galerkin projection of the Schrodinger operator onto the space spanned by solutions at these sample frequencies. The ROM matrix is in general full, and not good for extracting the potential. However, using an orthogonal change of basis via Lanczos iteration, we can transform the ROM to a block triadiagonal form from which it is easier to extract $q$. In one dimension, the tridiagonal matrix corresponds to a three-point staggered finite-difference system for the Schrodinger operator discretized on a so-called spectrally matched grid which is almost independent of the medium. In higher dimensions, the orthogonalized basis functions play the role of the grid steps. The orthogonalized basis functions are localized and also depend only very weakly on the medium, and thus by embedding into the continuous problem, the reduced order model yields highly accurate internal solutions. That is to say, we can obtain, just from boundary data, very good approximations of the solution of the Schrodinger equation in the whole domain for a spectral interval that includes the sample frequencies. We present inversion experiments based on the internal solutions in one and two dimensions.

中文翻译：

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谱域反演的降阶模型：嵌入到连续问题中并生成内部数据。

我们生成数据驱动的降阶模型 (ROM)，用于在给定几个频率的边界数据的谱域中对一维和二维薛定谔方程进行反演。ROM 是薛定谔算子在这些采样频率下解所跨越的空间上的伽辽金投影。ROM矩阵一般是满的，不利于挖掘潜力。然而，通过 Lanczos 迭代使用基的正交变化，我们可以将 ROM 转换为块三对角形式，从中更容易提取 $q$。在一维中，三对角矩阵对应于薛定谔算子的三点交错有限差分系统，该系统在所谓的光谱匹配网格上离散，该网格几乎与介质无关。在更高的维度上，正交化的基函数起到网格步长的作用。正交化的基函数是局部化的，并且仅非常弱地依赖于介质，因此通过嵌入到连续问题中，降阶模型产生高度准确的内部解。也就是说，我们可以仅从边界数据中获得薛定谔方程在包括采样频率的频谱间隔的整个域中的解的非常好的近似值。我们展示了基于一维和二维内部解的反演实验。仅从边界数据来看，对于包括采样频率的频谱间隔，薛定谔方程在整个域中的解的非常好的近似。我们展示了基于一维和二维内部解的反演实验。仅从边界数据来看，对于包括采样频率的频谱间隔，薛定谔方程在整个域中的解的非常好的近似。我们展示了基于一维和二维内部解的反演实验。