Weak Monge-Ampère solutions of the semi-discrete optimal transportation problem

Document Type

Syllabus

Publication Date

8-7-2017

Abstract

We consider the Monge-Kantorovich optimal transportation (OT) problem between two measures, one of which is a weighted sum of Diracs. This problem is traditionally solved using geometric methods based on the computation of Laguerre cells. We review the duality between Brenier/Pogorelov weak solutions and the classical Aleksandrov measure formulation. It is well known that the OT problem can be reformulated as a Monge-Ampère elliptic partial differential equation. However, existing numerical methods for this non-linear PDE require the measures to have finite density. We propose 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 discretization of the equation. Computational results demonstrate the correctness of this scheme when methods designed for conventional viscosity solutions fail.

Identifier

85093616219 (Scopus)

ISBN

[9783110439267, 9783110430417]

Publication Title

Topological Optimization and Optimal Transport in the Applied Sciences

External Full Text Location

https://doi.org/10.1515/9783110430417-009

First Page

175

Last Page

203

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