A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data

We consider the hybrid problem of reconstructing the isotropic electric conductivity of a body Ω from interior current density imaging data obtainable using MRI measurements. We only require knowledge of the magnitude |J| of one current for a given voltage f on the boundary ∂Ω. As previously shown, the corresponding voltage potential u in Ω is a minimizer of the weighted least gradient problem with a(x) = |J(x)|. In this paper, we present an alternating split Bregman algorithm for treating such least gradient problems, for a ∈ L2(Ω) non-negative and f ∈ H1/2(∂Ω). We give a detailed convergence proof by focusing to a large extent on the dual problem. This leads naturally to the alternating split Bregman algorithm (which is a re-interpretation of the alternating direction method of multipliers adapted to L1 problems). The dual problem also turns out to yield a novel method to recover the full vector field J from knowledge of its magnitude and of the voltage f on the boundary. We then present several numerical experiments that illustrate the convergence behavior of the proposed algorithm. (paper)