The ability to image and perform nondestructive testing of materials using electromagnetic, sound, or elastic waves is essential in many areas of national importance, such as the design and manufacturing of exotic materials, public safety, medical imaging, and underground exploration. Unfortunately, effective methods for testing complicated materials for structural imperfections or for identifying unknown targets with little a priori information are still in a state of infancy. In this NSF-funded project, the PI and her graduate students will develop entirely 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. This study will combine practical applications with the mathematical elegance of new imaging techniques that have recently led to the establishment of a new field in mathematics called 'qualitative methods.'Until not long ago, essentially all existing algorithms for target identification in inverse scattering theory were based on either a weak scattering approximation or on the use of nonlinear optimization techniques. Alternative methods for imaging have been developed to avoid incorrect model assumptions inherent in weak scattering approximations and the requirement of a priori information on the topological and physical properties of the unknown object. Such methods come under the general title of qualitative methods in inverse scattering theory, otherwise referred to as non-iterative methods. Roughly speaking, these methods explore the properties of a linear compact operator determined by the scattering data, which encodes nonlinear information about the scattering object and provides easily implementable reconstruction algorithms. Inherent to the methods appears a new eigenvalue problem, known as the transmission eigenvalue problem. The practical importance of transmission eigenvalues is that they can be determined from the measured scattering data and they carry information about the material properties of the scatterer. This proposal concerns broadening the applicability of the qualitative approach for solving inverse scattering problems with particular emphasis on inhomogeneous media. Of main concern are potential applications to nondestructive testing and target identification. The investigation will take two directions: 1) develop qualitative inversion approaches for inhomogeneous media containing small parameter features that are not in the Born approximation regime, 2) reduce the amount of spatial data needed for the qualitative approach by using only quasi-backscattering and time domain data. To accomplish the above objectives, the investigator and her graduate students will need to investigate new theoretical questions related to the (non-selfadjoint and non-linear) transmission eigenvalue problems that arise in the investigation of the above two parts.
|Effective start/end date||8/1/15 → 7/31/18|
- National Science Foundation (National Science Foundation (NSF))