First-order System Least-squares for Electrical Impedance Tomography

Hugh R. MacMillan


The practical limitations of electrical impedance tomography (EIT), due to both the necessarily finite set of inexact boundary data and the diffusive nature of the current into the interior, lead to the conclusion that reconstructing the interior impedance is an ill-posed problem. Given a set of applied normal boundary currents, a standard approach to EIT is to minimize the defect between the known and the computed boundary voltages that are associated, respectively, with the exact impedance and its approximation. In minimizing a boundary functional, standard least-squares approaches implicitly impose the interior partial differential equation (PDE), the diffusion equation. We have developed and implemented a first-order system least-squares (FOSLS) formulation which incorporates the elliptic PDE in a global multigrid minimization scheme. To place the new functional in context, we establish its equivalence to existing least-squares approaches, and to a novel norm on the error in the approximate impedance. The effect of this equivalence is to guarantee that, for each formulation, there is a unique minimizer in the topology corresponding to this special norm. Thus, the theory quantifies the sense in which the fully nonlinear inverse problem is well posed, and guides the discretization according to those components which we can expect to reconstruct, regardless of the computational objective. With this framework established, {\it a priori} information, which might otherwise be used to regularize, can be incorporated by introducing an additional term to the functional.