Anisotropic inverse problems with internal measurements

This thesis concerns the hybrid inverse problem of reconstructing a tensor-valued conductivity from knowledge of internal measurements. This problem finds applications in the medical imaging modalities Current Density Imaging and Magnetic Resonance Electrical Impedance Tomography. In the first part of the thesis, we investigate the reconstruction of the anisotropic conductivity in a second-order elliptic partial differential equation, from knowledge of internal current densities. We show that the unknown coefficient can be uniquely and stably reconstructed via explicit inversion formulas with a loss of one derivative compared to errors in the measurement. This improves the resolution of quantitative reconstructions in Calderón's problem (i.e. reconstruction problems from knowledge of boundary measurements). We then extend the problem to the full anisotropic Maxwell system and show that the complex-valued anisotropic admittivity can be uniquely reconstructed from knowledge of several internal magnetic fields. We also proved a unique continuation property and Runge approximation property for an anisotropic Maxwell system. In the second part, we performed some numerical experiments to demonstrate the computational feasibility of the reconstruction algorithms and assess their robustness to noisy measurements.