Scaling limits of discrete optimal transport
P. Gladbach, E. Kopfer, J. Maas, SIAM Journal on Mathematical Analysis 52 (2020) 2759โ2802.
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Author
Gladbach, Peter;
Kopfer, Eva;
Maas, JanIST Austria 

Department
Abstract
We consider dynamical transport metrics for probability measures on discretisations of a bounded convex domain in โd. These metrics are natural discrete counterparts to the Kantorovich metric ๐2, defined using a Benamou-Brenier type formula. Under mild assumptions we prove an asymptotic upper bound for the discrete transport metric Wt in terms of ๐2, as the size of the mesh T tends to 0. However, we show that the corresponding lower bound may fail in general, even on certain one-dimensional and symmetric two-dimensional meshes. In addition, we show that the asymptotic lower bound holds under an isotropy assumption on the mesh, which turns out to be essentially necessary. This assumption is satisfied, e.g., for tilings by convex regular polygons, and it implies Gromov-Hausdorff convergence of the transport metric.
Publishing Year
Date Published
2020-10-01
Journal Title
SIAM Journal on Mathematical Analysis
Volume
52
Issue
3
Page
2759-2802
ISSN
eISSN
IST-REx-ID
Cite this
Gladbach P, Kopfer E, Maas J. Scaling limits of discrete optimal transport. SIAM Journal on Mathematical Analysis. 2020;52(3):2759-2802. doi:10.1137/19M1243440
Gladbach, P., Kopfer, E., & Maas, J. (2020). Scaling limits of discrete optimal transport. SIAM Journal on Mathematical Analysis. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/19M1243440
Gladbach, Peter, Eva Kopfer, and Jan Maas. โScaling Limits of Discrete Optimal Transport.โ SIAM Journal on Mathematical Analysis. Society for Industrial and Applied Mathematics, 2020. https://doi.org/10.1137/19M1243440.
P. Gladbach, E. Kopfer, and J. Maas, โScaling limits of discrete optimal transport,โ SIAM Journal on Mathematical Analysis, vol. 52, no. 3. Society for Industrial and Applied Mathematics, pp. 2759โ2802, 2020.
Gladbach P, Kopfer E, Maas J. 2020. Scaling limits of discrete optimal transport. SIAM Journal on Mathematical Analysis. 52(3), 2759โ2802.
Gladbach, Peter, et al. โScaling Limits of Discrete Optimal Transport.โ SIAM Journal on Mathematical Analysis, vol. 52, no. 3, Society for Industrial and Applied Mathematics, 2020, pp. 2759โ802, doi:10.1137/19M1243440.
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arXiv 1809.01092