Inverse Scattering Using Data-Driven Reduced-Order Models

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Waves are used to image in many applications like radar, ultrasound or geophysical exploration. This project is concerned with developing a novel reduced-order model based methodology for inverse scattering and for imaging of periodic, dispersive structures in nano-photonics. Reduced-order models are extensively used in the dynamical-systems community; however, their use in inverse problems has been limited. The inverse scattering problems in this project are formulated for typical setups, where data are gathered by arrays of sensors that probe an unknown heterogeneous medium with waves and measure the returns. Rather than asking the question "which scattering medium could have caused the measured wave returns" directly, this novel methodology first constructs a reduced-order model from the measured data. The reduced-order model is defined to preserve the physics of wave propagation. The research is motivated by applications in nondestructive evaluation of materials, estimation of sea ice thickness using airborne radar, medical imaging, oil exploration and nano-photonics. Open source software will be provided in the scope of this project This project seeks new directions in inverse scattering and reduced-order modeling. It shows how to view wave propagation as a dynamical system driven by a linear operator that contains information about the medium, modeled by unknown coefficients in the acoustic or electromagnetic wave equation. The dynamical system framework allows the connection to reduced-order modeling techniques, which need to be adapted in order to be useful in inverse scattering. The project pursues this goal for the following specific questions: (1) How can one control waves for improved focusing and subsequent improved imaging, in a pixel scanning fashion? (2) How can one construct and use a ROM for quantitative estimation of the coefficients in the governing wave equations? (3) How can one invert in lossy media? The latter question is connected to an important problem in reduced-order modeling, concerned with the identification of port-Hamiltonian dynamical systems from measurements of their transfer function. This project also seeks to develop model-driven ROMs for dispersive, periodic nano-photonic structures, for speeding up design optimization and parametric inversion.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.