Date of Award
Doctor of Philosophy (PhD)
Computational Analysis and Modeling
Direct and inverse scattering problems have wide applications in geographical exploration, radar, sonar, medical imaging and non-destructive testing. In many applications, the obstacles are not smooth. Corner singularity challenges the design of a forward solver. Also, the nonlinearity and ill-posedness of the inverse problem challenge the design of an efficient, robust and accurate imaging method.
This dissertation presents numerical methods for solving the direct and inverse scattering problems for domains with multiple corners. The acoustic wave is sent from the transducers, scattered by obstacles and received by the transducers. This forms the response matrix data. The goal for the direct scattering problem is to compute the response matrix data using the knowledge of the shape of the obstacles. The goal for the inverse scattering problem is to image the location and geometry of the obstacles based on the response matrix data. Both the near field and far field cases are considered. For the direct problem, the challenges of logarithmic singularity from Green's functions and corner singularity are both taken care of. For the inverse problem, an efficient and robust direct imaging method similar to the Multiple Signal Classification algorithm is proposed. Multiple frequency data are combined to capture details while not losing robustness. The near field and far field response matrices are compared and their singular value patterns are compared as well. The singular value perturbation is carefully studied. Extensive numerical results demonstrate that our forward solver is capable of handling domains with multiple corners by solving a linear system with low condition numbers generated from a boundary integral equation, that our inverse problem solver is efficient, accurate and robust. It could handle response matrix data with noise.
Yihong, Jiang, "" (2016). Dissertation. 114.