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arXiv · 1905.05757

Homogenisation of one-dimensional discrete optimal transport

Peter Gladbach·Eva Kopfer·Jan Maas·Lorenzo Portinale
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Abstract

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.

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Peter Gladbach, Eva Kopfer, Jan Maas, Lorenzo Portinale. 2020-01-23. Homogenisation of one-dimensional discrete optimal transport. https://arxiv.org/abs/1905.05757

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