  We consider the Monge-Kantorovich optimal transportation problem between two
measures, one of which is a weighted sum of Diracs. This problem is
traditionally solved using expensive geometric methods. It can also be
reformulated as an elliptic partial differential equation known as the
Monge-Ampere equation. However, existing numerical methods for this non-linear
PDE require the measures to have finite density. We introduce a new formulation
that couples the viscosity and Aleksandrov solution definitions and show that
it is equivalent to the original problem. Moreover, we describe a local
reformulation of the subgradient measure at the Diracs, which makes use of
one-sided directional derivatives. This leads to a consistent, monotone
discretisation of the equation. Computational results demonstrate the
correctness of this scheme when methods designed for conventional viscosity
solutions fail.
