  We propose and study a regularization method for recovering an approximate
electrical conductivity solely from the magnitude of one interior current
density field. Without some minimal knowledge of the boundary voltage
potential, the problem has been recently shown to have nonunique solutions,
thus recovering the exact conductivity is impossible. The method is based on
solving a weighted least gradient problem in the subspace of functions of
bounded variations with square integrable traces. The computational
effectiveness of this method is demonstrated in numerical experiments.
