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

TL;DR

This paper introduces a viscosity-based framework for solving Pogorelov solutions of the Monge-Ampere equation, enabling efficient numerical computation for optimal transport problems involving measures with Dirac components.

Contribution

It develops a new formulation coupling viscosity and Aleksandrov solutions, with a local reformulation for Dirac measures, leading to a consistent, monotone discretisation that improves computational robustness.

Findings

The scheme correctly computes Pogorelov solutions where traditional viscosity methods fail.

The formulation is equivalent to the original optimal transport problem with Dirac measures.

Numerical results validate the effectiveness of the proposed discretisation.

Abstract

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.