Calculus of Variations and Geometric Measure Theory

P. Gladbach - E. Kopfer - J. Maas - L. Portinale

Homogenisation of one-dimensional discrete optimal transport

created by portinale on 22 Jun 2021
modified on 05 Oct 2022


Published Paper

Inserted: 22 jun 2021
Last Updated: 5 oct 2022

Journal: Journal des Mathématiques Pures et Appliquées
Year: 2020

ArXiv: 1905.05757 PDF


This paper deals with dynamical optimal transport metrics defined by spatial discretisation of the Benamou--Benamou formula for the Kantorovich metric $W_2$. Such metrics appear naturally in discretisations of $W_2$-gradient flow formulations for dissipative PDE. However, it has recently been shown that these metrics do not in general converge to $W_2$, unless strong geometric constraints are imposed on the discrete mesh. In this paper we prove that, in a $1$-dimensional periodic setting, discrete transport metrics converge to a limiting transport metric with a non-trivial effective mobility. This mobility depends sensitively on the geometry of the mesh and on the non-local mobility at the discrete level. Our result quantifies to what extent discrete transport can make use of microstructure in the mesh to reduce the cost of transport.