Scaling limits of discrete optimal transport

P. Gladbach, E. Kopfer, J. Maas, SIAM Journal on Mathematical Analysis 52 (2020) 2759–2802.


Journal Article | Published | English

Scopus indexed
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 ℝd. These metrics are natural discrete counterparts to the Kantorovich metric π•Ž2, 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 π•Ž2, 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
71

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, 52(3), 2759–2802. https://doi.org/10.1137/19M1243440
Gladbach, Peter, Eva Kopfer, and Jan Maas. β€œScaling Limits of Discrete Optimal Transport.” SIAM Journal on Mathematical Analysis 52, no. 3 (2020): 2759–2802. https://doi.org/10.1137/19M1243440.
P. Gladbach, E. Kopfer, and J. Maas, β€œScaling limits of discrete optimal transport,” SIAM Journal on Mathematical Analysis, vol. 52, no. 3, pp. 2759–2802, 2020.
Gladbach P, Kopfer E, Maas J. 2020. Scaling limits of discrete optimal transport. SIAM Journal on Mathematical Analysis. 52(3), 2759–2802.
Gladbach, Peter, et al. β€œScaling Limits of Discrete Optimal Transport.” SIAM Journal on Mathematical Analysis, vol. 52, no. 3, Society for Industrial and Applied Mathematics, 2020, pp. 2759–802, doi:10.1137/19M1243440.
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