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We combine data-driven reduced order models (ROM) with the Lippmann- Schwinger integral equation to produce a direct nonlinear inversion method. The ROM is viewed as a Galerkin projection and is sparse due to Lanczos orthogonalization. Embedding into the continuous problem, a data-driven internal solution is produced. This internal solution is then used in the Lippmann-Schwinger equation, in a direct or iterative framework. The new approach also allows us to process non-square matrix-valued data-transfer functions, i.e., to remove the main limitation of the earlier versions of the ROM based inversion algorithms. We show numerical experiments for spectral domain data for which our inversion is far superior to the Born inversion.
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On extension of the data driven ROM inverse scattering framework to partially nonreciprocal arrays
Abstract Data-driven reduced order models (ROMs) recently emerged as powerful tool for the solution of inverse scattering problems. The main drawback of this approach is that it was limited to measurement arrays with reciprocally collocated transmitters and receivers, that is, square symmetric matrix (data) transfer functions. To relax this limitation, we use our previous work Druskin et al (2021 Inverse Problems 37 075003), where the ROMs were combined with the Lippmann–Schwinger integral equation to produce a direct nonlinear inversion method. In this work we extend this approach to more general transfer functions, including those that are non-symmetric, e.g., obtained by adding only receivers or sources. The ROM is constructed based on the symmetric subset of the data and is used to construct all internal solutions. Remaining receivers are then used directly in the Lippmann–Schwinger equation. We demonstrate the new approach on a number of 1D and 2D examples with non-reciprocal arrays, including a single input/multiple outputs inverse problem, where the data is given by just a single-row matrix transfer function. This allows us to approach the flexibility of the Born approximation in terms of acceptable measurement arrays; at the same time significantly improving the quality of the inversion compared to the latter for strongly nonlinear scattering effects.
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- PAR ID:
- 10340863
- Date Published:
- Journal Name:
- Inverse Problems
- Volume:
- 38
- Issue:
- 8
- ISSN:
- 0266-5611
- Page Range / eLocation ID:
- 084002
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1088/1361-6420/ac7a59
