Applications of microlocal analysis to some hyperbolic inverse problems
Abstract
This thesis compiles my work on three inverse problems: ultrasound recovery in thermoacoustic tomography, cancellation of singularities in synthetic aperture radar, and the injectivity and stability of some generalized Radon transforms. Each problem is approached using microlocal methods. In the context of thermoacoustic tomography under the damped wave equation, I show uniqueness and stability of the problem with complete data, provide a reconstruction algorithm for small attenuation with complete data, and obtain stability estimates for visible singularities with partial data. The chapter on synthetic aperture radar constructs microlocally several infinite-dimensional families of ground reflectivity functions which appear microlocally regular when imaged using synthetic aperture radar. Finally, based on a joint work with Hanming Zhou, we show the analytic microlocal regularity of a class of analytic generalized Radon transforms, using this to show injectivity and stability for a generic class of generalized Radon transforms defined on analytic Riemannian manifolds.
Degree
Ph.D.
Advisors
Stefanov, Purdue University.
Subject Area
Mathematics
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