First-order system least squares and electrical impedance tomography

Electrical impedance tomography (EIT) belongs to a family of imaging methods that employ boundary measurements to distinguish interior spatial variation of an electromagnetic parameter. The associated inverse problem is notoriously ill-posed, due to diffusive effects in the quasi-static regime, when electrical impedance reduces to its real part, resistivity. The standard approach to EIT is output least squares (OLS). For a set of applied normal boundary currents, one minimizes the defect between the measured and computed boundary voltages associated, respectively, with the exact impedance and its approximation. In minimizing a boundary functional, OLS implicitly imposes the governing Poisson equation as an optimization constraint. To reconstruct resistivity or, equivalently, conductivity, we introduce a new first-order system least-squares (FOSLS) formulation that incorporates the elliptic PDE as an interior functional in a global unconstrained minimization scheme. We then establish equivalence of our functional to OLS and to an existing interior least-squares functional due to Kohn and Vogelius [Comm. Pure Appl. Math, 37 ( 1984), pp. 289-298]. That the latter may be viewed as a special dual approach (FOSLL*) is an interesting attribute of this equivalence. Because the FOSLS functional implicitly reflects the inherent loss of resolution away from the boundary, relative to the set of applied boundary tests, the need for artificial regularization may be avoided.