Well-posedness in anisotropic conductivity reconstruction

Abstract

In the thesis, interior conductivity reconstruction problems inferring from the interior current density data are considered. It is our main result that an anisotropic conductivity reconstruction in two dimensions is well-posed. This is relevant to an application of a medical diagnosis on human organism since human organism typically has an anisotropic conductivity.

In the first part of thesis, the well-posedness theorems on isotropic, orthotropic, and anisotropic conductivities are presented. An existence theorem is treated to the same extent as an uniqueness theorem; The admissibility conditions on the current data that are sufficient to imply the existence of the solution are defined prior to stating the theorems. They provide sufficient conditions to characterize differences between arbitrary vector fields and the current vector fields that are realizable electrically. The main results comes from that we can pose an equation or a system of equations of hyperbolic type on the solution.

In the second part of thesis, a numerical algorithm to reconstruct the conductivity is suggested. Isotropic and orthotropic materials can be treated by the algorithm. The algorithm has two advantages: Firstly, the conductivity reconstruction is obtained by directly solving equations of hyperbolic type without iteration process. Secondly, numerical approximations of divergence and curl operator are exact, which is known to be of mimetic type. Since this scheme is realized as a resistive network of circuit theory in our context, we call the algorithm Virtual Resistive Network algorithm. Propagation of noises and the stability of the algorithm is investigated. Due to the nature of the hyperbolic problems, the noises would propagate along characteristic lines without cancellation. The numerical scheme composed by the mimetic idea turns out to be the one that may relax the hyperbolic nature of noise propagation.