Unnormalized optimal transport

We propose an extension of the computational fluid mechanics approach to the Monge-Kantorovich mass transfer problem, which was developed by Benamou-Brenier in [4]. Our extension allows optimal transfer of unnormalized and unequal masses. We obtain a one-parameter family of simple modifications of the formulation in [4]. This leads us to a new Monge-Ampére type equation and a new Kantorovich duality formula. These can be solved efficiently by, for example, the Chambolle-Pock primal-dual algorithm [6]. This solution to the extended mass transfer problem gives us a simple metric for computing the distance between two unnormalized densities. The L1 version of this metric was shown in [25] (which is a precursor of our work here) to have desirable properties.