  We propose an extension of the computational fluid mechanics approach to the
Monge-Kantorovich mass transfer problem, which was developed by
Benamou-Brenier. 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-Ampere type equation and a new
Kantorovich duality formula. These can be solved efficiently by, for example,
the Chambolle-Pock primal-dual algorithm. 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 [23]
(which is a precursor of our work here) to have desirable properties.
