  We propose a fast algorithm to approximate the optimal transport distance.
The main idea is to add a Fisher information regularization into the dynamical
setting of the problem, originated by Benamou and Brenier. The regularized
problem is shown to be smooth and strictly convex, thus many classical fast
algorithms are available. In this paper, we adopt Newton's method, which
converges to the minimizer with a quadratic rate. Several numerical examples
are provided.
