Inverse Problems for Wave Phenomena

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This mathematics research project by Plamen Stefanov concerns the study of inverse problems arising in wave propagation phenomena. They include the following: (a) inverse problems in integral geometry of recovery a function or a tensor field from the geodesic ray transform; (b) microlocal inversion of transforms appearing in Synthetic Aperture Radar (SAR) imaging; (c) non-linear boundary and lens rigidity questions in Riemannian geometry; (d) inverse boundary problems for wave equations with variable speed; and (e) inverse problems for the wave equation arising in multiwave medical imaging. In (a), (c), and (d), we assume that there are conjugate points and the goal is to study their effect on the invertibility and its stability. The unifying theme in the project is that the inversion is done in complex geometry. A straightforward inversion in all those cases fails due to the fact that the microlocal Bolker condition does not hold anymore. One often gets artifacts in the reconstruction placed at conjugate pairs or at mirror points in SAR, etc. The fundamental questions that Stefanov will study are whether those artifacts are indeed there; how to remove them, if possible; and, (if not possible) how to characterize them. The long term goal is to understand linear and non-linear inverse problems and their stability for wave phenomena in complex media, where caustics are inevitable and stability might be lost. The work on those problems is based upon, and will further develop, methods in several areas in mathematics: Riemannian and Integral geometry, Inverse Problems, microlocal analysis, including analytic microlocal analysis, as well as classical functional analysis and PDE theory. This mathematics research project by Plemen Stefanov is motivated by problems arising in Synthetic Aperture Radar imaging (SAR), medical imaging and geophysics. In SAR, Stefanov will study practical questions such as what are the best flight paths that produce the best data for image reconstruction; Stefanov will also develop an approximate reconstruction algorithm for situations where there is no exact answer to this question. The boundary and the lens rigidity problems studied in this project are the mathematical foundation of Seismic Tomography: to recover the inner structure of the Earth (and the Sun) from the travel times of seismic waves; they also occur in the investigation of ultrasound imaging. The geodesic ray transform problems are a special case (linearization) of the latter. The inverse wave equation problems arise in Seismology as well, as an attempt to use more information about the wave -- not just the time of arrival but also the shape of the wave. The multiwave problems are the mathematical foundation of emerging medical imaging methods like Thermoacoustic and Photoacoustic Tomography; they belong to the new class of multiwave (hybrid) medical imaging methods which combine the high contrast of one wave with the high resolution of another.