Inverse problems and microlocal analysis in various regularity classes

In this thesis we investigate three different inverse problems arising from hyperbolic and elliptic partial differential equations. The thread connecting these is the use of microlocal analytic techniques in various regularity classes as a common tool set -- these classes are real-analytic, Gevrey and smooth. In particular, we show that the inverse problem of recovering a time-independent potential from the source-to-solution map of the wave equation is exponentially unstable in some domain. We demonstrate that the perturbation of some sound speed of waves travelling underground can be recovered uniquely under some geometric and analytic assumptions. Finally, we prove that it is possible to uniquely recover an unbounded potential in the Schrödinger equation from partial Neumann-to-Dirichlet data.