Primal and Dual Problems in Electrical Impedance Imaging

III-posed inverse boundary value problems are usually approached as problems of inference under a given set of boundary observations. The parameters and data are modelled as random variables so that to encompass the uncertainty of their actual values. This uncertainty is expressed in probability distributions of the variables which are subsequently conjuncted to form the basis of the regularized inverse problem. In this work, we consider the linearized inverse conductivity problem, also known as electrical impedance imaging problem, where a finite set of noise infused boundary voltage measurements is used to reconstruct the conductivity distribution in the interior of simply connected domains. We derive the primal and dual problems in the generalized Tikhonov formulation, and cast the linearized inverse problem as a quadratic optimization problem with an inequality l2norm constraint. We show that Tikhonov regularization can be implemented in the context of primal-dual interior point methods to yield optimal images and regularization parameters with respect to the choice of prior information equations and the noise level in the data. The approach can be extended in nonlinear trust-region regularization using algorithms based on consecutive linearization steps.