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Title: Lippmann-Schwinger-Lanczos algorithm for inverse scattering problems
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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2022 Spring Central Sectional Meeting
Sponsoring Org:
National Science Foundation
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  1. 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 themore »latter for strongly nonlinear scattering effects.« less
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In his seminal 1950s works Krein realized that in addition to their compressing properties, network realizations can be used to embed the data back into the state space of the underlying continuum problems. In more recent works of the authors Krein's ideas gave rise to so-called nite-dierence Gaussian quadrature rules (FDGQR), allowing to approximately map the ROM state-space representation to its full order continuum counterpart on a judicially chosen grid. Thus, the state variables can be accessed directly from themore »transfer function without solving the full problem and even explicit knowledge of the PDE coecients in the interior, i.e., the FDGQR directly learns" the problem from its transfer function. This embedding property found applications in PDE solvers, inverse problems and unsupervised machine learning. Here we show a generalization of this approach to dissipative PDE problems, e.g., electromagnetic and acoustic wave propagation in lossy dispersive media. 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A two-point positive denite problem in a semi-innite domain, SIAM Journal on Numerical Analysis, V. 37, N 2, pp.403422, 1999 [3] Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Distance preserving model order reduction of graph-Laplacians and cluster analysis, Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Journal of Scientic Computing, V. 90, N 1, pp 130, 2022 [4] Druskin, Vladimir and Moskow, Shari and Zaslavsky, Mikhail LippmannSchwingerLanczos algorithm for inverse scattering problems, Inverse Problems, V. 37, N. 7, 2021, [5] Mark Adolfovich Nudelman The Krein String and Characteristic Functions of Maximal Dissipative Operators, Journal of Mathematical Sciences, 2004, V 124, pp 49184934 Go back to Plenary Speakers Go back to Speakers Go back« less
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