| This thesis studies a scale of hybrid inverse problems in noninvasive imaging, also called two-physics problems. The main focus is on recovering an unknown conductivity inside the body, gamma(x), which is an unknown coefficient governing an electrical potential, u(x). The potential satisfies the conductivity equation with a known Dirichlet boundary condition. The hybrid problem assumes access to an internal functional, F(gamma( x), u(x), ▿u( x)), which depends on the conductivity, the electrical potential and its gradient. In many instances it takes the form F( x; gamma) = gamma|▿u(x;gamma)| p, x ∈ O. When p = 1, we are able to recover the conductivity numerically in MATLAB, using the J-substitution method by Kwon et al. [2002]. Simulations are performed on various phantoms, which are conductivities with jumps across various interfaces. A second goal is to prove that the linearization of the nonlinear map from the conductivity to the internal functional is Fredholm. The analysis is carried out completely in the one-dimensional case. For two- and three- dimensions, such results are known for smooth background conductivities from the work of Kuchment and Steinhauer [2012], using microlocal analysis. The goal of this research is to understand what happens in the more physically realistic case of non-smooth background conductivities. The conductivity is assumed to be in a conormal class, allowing for jumps in derivatives. We use techniques from microlocal analysis, including Fourier integral operators and Ip,l classes of Fourier integral distributions associated to two intersecting Lagrangian manifolds. We also obtain a Green's function for the conductivity operator. |
