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 the knowledge of the magnitude |J| of one current generated by a given voltage f on the boundary ∂Ω. The corresponding voltage potential u in Ω is a minimizer of the weighted least gradient problem
u=argmin{∫Ωa(x)|∇u|:u∈H1(Ω), u|∂Ω=f}, with a(x)=|J(x)|. In this talk I will present an alternating split Bregman algorithm for treating such least gradient problems, for a∈L2(Ω) non-negative and f∈H1/2(∂Ω).
I will sketch a convergence proof by focusing to a large extent on the dual problem. This leads naturally to the alternating split Bregman algorithm. 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. I will present several numerical experiments that illustrate the convergence behavior of the proposed algorithm. This is a joint work with A. Nachman and A. Timonov.