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Data-driven reduced order models (ROMs) have recently emerged as an efcient tool for the solution of inverse scattering problems with applications to seismic and sonar imaging. One requirement of this approach is that it uses the full square multiple-input/multiple-output (MIMO) matrixvalued transfer function as the data for multidimensional problems. The synthetic aperture radar (SAR), however, is limited to the single-input/single-output (SISO) measurements corresponding to the diagonal of the matrix transfer function. Here we present a ROM-based Lippmann-Schwinger approach overcoming this drawback. The ROMs are constructed to match the data for each source-receiver pair separately, and these are used to construct internal solutions for the corresponding source using only the data-driven Gramian. Efficiency of the proposed approach is demonstrated on 2D and 2.5D (3D propagation and 2D reflectors) numerical examples. The new algorithm not only suppresses multiple echoes seen in the Born imaging but also takes advantage of their illumination of some back sides of the reflectors, improving the quality of their mapping. 

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 callKrein 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. 

Abstract Rational approximation recently emerged as an efficient numerical tool for the solution of exterior wave propagation problems. Currently, this technique is limited to wave media which are invariant along the main propagation direction. We propose a new model order reduction-based approach for compressing unbounded waveguides with layered inclusions. It is based on the solution of a nonlinear rational least squares problem using the RKFIT method. We show that approximants can be converted into an accurate finite difference representation within a rational Krylov framework. Numerical experiments indicate that RKFIT computes more accurate grids than previous analytic approaches and even works in the presence of pronounced scattering resonances. Spectral adaptation effects allow for finite difference grids with dimensions near or even below the Nyquist limit.