## Abstract

We present a reduced-order model (ROM) methodology for inverse scattering problems in which the ROMs 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.

Original language | English |
---|---|

Article number | 025002 |

Number of pages | 25 |

Journal | Inverse Problems |

Volume | 40 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2024 |

### Funding

We thank Liliana Borcea, Alex Mamonov, and Mikhail Zaslavsky for many stimulating discussions. The work of Vladimir Druskin was financially supported by AFOSR Grants FA 955020-1-0079 and FA9550-23-1-0220, and NSF Grant DMS-2110773. The work of Elena Cherkaev was financially supported by NSF Grant DMS-2111117. The work of Murthy Guddati was financially supported by NSF Grant DMS-2111234. The work of Jörn Zimmerling was financially supported by NSF Grant DMS-2110265. This support is gratefully acknowledged.## Keywords

- embedding
- inverse scattering
- optimal grids
- reduced-order models