Stability and regularity of an inverse elliptic boundary value problem
Dobson, David Clark
Symes, William W.
Doctor of Philosophy
The inverse conductivity problem is that of recovering a spatially varying isotropic conductivity in the interior of some bounded region by means of steady-state measurements taken only at the boundary. In the underlying partial differential equation, the conductivity appears as a coefficient in an elliptic boundary value problem. We first analyze the stability of the formal linearization of the inverse conductivity problem, establishing upper and lower bounds on the linearized map. Conditions are then established under which the forward map is regular, with computationally reasonable norms on the conductivity and the data. Certain smoothness assumptions on the conductivities are needed to prove regularity. A simple example is given to illustrate why the smoothness assumptions may be necessary. Finally, the inverse problem is formulated as a regularized minimization problem. The regularization penalizes rough conductivities, rendering the forward map regular and stabilizing the linearized inverse maps. The local convergence of a simple minimization scheme is established.