Abstract Datadriven 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 nonsymmetric, 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 nonreciprocal arrays, including a single input/multiple outputs inverse problem, where the data is given by just a singlerow 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 themore »
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LippmannSchwingerLanczos algorithm for inverse scattering problems
We combine datadriven 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 datadriven internal solution is produced. This internal solution is then used in the LippmannSchwinger equation, in a direct or iterative framework. The new approach also allows us to process nonsquare matrixvalued datatransfer 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.
 Award ID(s):
 2110773
 Publication Date:
 NSFPAR ID:
 10340872
 Journal Name:
 2022 Spring Central Sectional Meeting
 Sponsoring Org:
 National Science Foundation
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