In many areas of national importance, such as the design and manufacturing of exotic materials, public safety, medical imaging, and underground exploration, it is essential to be able to image and perform nondestructive testing of materials using electromagnetic, sound, or elastic waves. Unfortunately, effective methods for testing complicated materials for structural defects or for identifying unknown targets in an efficient way with little a priori information are still in a state of infancy. In this project, the investigator and her graduate students will develop new techniques in inverse scattering theory to obtain reliable target signatures or usable information about objects being examined in computationally efficient ways. The goal is to minimize dependence on a priori information describing the physics and/or geometry of unknown targets as well as of the complications arising from complexity of the hosting background. This study will combine practical applications with the mathematical elegance of new direct imaging techniques. This research is a multifaceted effort to investigate some open questions associated with the qualitative methods (otherwise referred to as non-iterative methods) for solving inverse scattering problems for inhomogeneous media. The central theme is the development of the generalized linear sampling method for a variety of inverse problems. This is a new qualitative approach to inverse scattering that is rigorously justified for noisy data. There are three main projects: 1) Eigenvalues in inverse scattering theory for inhomogeneous media: Motivated by the theory of the transmission eigenvalue problem, this project proposes to develop a general framework for modifying the scattering operator in order to provide new eigenvalue problems associated with the scattering by an inhomogeneity. Particularly important are the issues of determining these (real or complex) eigenvalues from the scattering data, together with their relation to material properties of the inhomogeneity. The ultimate goal of this effort is to use such eigenvalues for the imaging of anisotropic (possibly absorbing and dispersive) media. 2) Qualitative methods for time domain problems: The use of time dependent data is proposed to address the issue of the need for large amount of spatial data in the use of qualitative reconstruction methods. The time domain interior transmission problem is the fundamental mathematical ingredient to develop qualitative methods for the wave equation and its solvability is a long lasting open problem. This project suggests an approach to solving this problem, hence opening the way to study time domain qualitative methods. 3) Inverse scattering for periodic media: The main consideration of this project is to investigate the far field behavior due to scattering by a highly oscillating periodic media of bounded support with application to imaging of macro/micro structure of the media. In addition, the project will develop a new qualitative method for reconstructing local perturbations inside periodic media without needing to compute the scattered field due to the periodic background nor to recover the background. The PI will undertake a systematic investigation of new qualitative methods in inverse scattering theory for complex inhomogeneous media as well as in the time domain which, if successful, can lead to progress outside the field of mathematics, such as nondestructive testing and target identification. In addition to progress outside the field of mathematics, the new eigenvalue problems that appear in this study, such as the transmission eigenvalue problem or the Steklov eigenvalue problem for absorbing media, have attracted considerable attention in the area of non-selfadjoint eigenvalue problems for partial differential equations. The analysis of scattering problems for periodic media touches some of the most active contemporary topics in PDEs.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.
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