Some Inverse Problems in Analysis and Geometry
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The aim of a typical inverse problem is to recover the interior properties of a medium by making measurements only on the boundary. These types of problems are motivated by geophysics, medical imaging and quantum mechanics among other fields. In this thesis, we consider two inverse problems arising in partial differential equations and geometry. The first part is devoted to the Calder\'on's problem with partial data. We consider the problem of developing a method to reconstruct a potential $q$ from the partial data Dirichlet-to-Neumann map for the Schr\"odinger equation $(-\Delta_g+q)u=0$ on a fixed admissible manifold $(M,g)$, where $\Delta_g$ is the Laplace-Beltrami operator. If the part of the boundary that is inaccessible for measurements satisfies a certain flatness condition in one direction, then we reconstruct the local attenuated geodesic ray transform of the one-dimensional Fourier transform of the potential $q$. This allows us to reconstruct $q$ locally, if the local (unattenuated) geodesic ray transform is constructively invertible. We also reconstruct $q$ globally, if $M$ satisfies certain concavity condition and if the global geodesic ray transform can be inverted constructively. In the second part, we study the Gaussian thermostat ray transforms on both closed Riemannian surfaces and compact Riemannian surfaces with boundary. We establish results on the injectivity of the thermostat ray transform, under certain conditions, and the surjectivity of its adjoint.
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