Section: New Results

A viscosity framework for computing Pogorelov solutions of the Monge-Ampere equation

Benamou, Jean-David and Froese, Brittany D.

[21]

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.

The method offers a new insight into the duality between Aleksandrov and Brenier solutions of the Monge Ampère equations. We still work on the viscosity existence/uniqueness convergence of sheme theory.