On Hybrid Inverse Problems for Partial Differential Equations

This thesis is concerned with the mathematical analysis of inverse problems for partial differential equations, and with a specific type of such problems known as hybrid inverse problems. Chapters 1-4 present various aspects of the theory of inverse problems for partial differential equations. Chapters 5-6 introduce the topic of hybrid inverse problems and content of the scientific papers in the appendix.

The scientific papers are concerned with the analysis of hybrid inverse problems. Specifically, we consider photoacoustic tomography and acousto-electric tomography, as well as the multi-frequency inverse source problem for the Helmholtz equation. For photoacoustic tomography, we analyze the stability and uniqueness of the inverse problem of reconstructing the optical coefficients of an object in a situation where measurement data is collected by a measurement device that can rotate around the object.

For acousto-electric tomography, we provide a method to reconstruct the electrical conductivity from boundary measurements of the voltage. Moreover, we investigate the feasibility of acousto-electric tomography for medical imaging by examining how reconstructions depend on the strength of the acousto-electric coupling.

In the case of multi-frequency measurements of solutions to the Helmholtzequation, we provide a detailed characterization of sources that can be stably reconstructed. The characterization depends on the measurement frequencies, and we devise an effective and stable reconstruction method.