We consider the hybrid problem of reconstructing the isotropic electric conductivity of a body $\Omega$ from the knowledge of the magnitude $|J|$ of one current generated by a given voltage $f$ on the boundary $\partial\Omega$. The corresponding voltage potential $u$ in $\Omega$ is a minimizer of the weighted least gradient problem

\[u=\hbox{argmin} \{\int_{\Omega}a(x)|\nabla u|: u \in H^{1}(\Omega), \ \ u|_{\partial \Omega}=f\},\] with $a(x)= |J(x)|$. In this talk I will present an alternating split Bregman algorithm for treating such least gradient problems, for $a\in L^2(\Omega)$ non-negative and $f\in H^{1/2}(\partial \Omega)$.

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.