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Transition to nestedness in multi- to one-dimensional optimal transport

  • PIERRE-ANDRÉ CHIAPPORI (a1), ROBERT MCCANN (a2) and BRENDAN PASS (a3)

Abstract

We study a one-parameter class of examples of optimal transport problems between a two-dimensional source and a one-dimensional target. Our earlier work identified a nestedness condition on the surplus function and marginals, under which it is possible to solve the problem semi-explicitly. In the family of examples we consider, we classify the values of parameters which lead to nestedness. In those cases, we derive an almost explicit characterisation of the solution.

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[1]Chiappori, P.-A., McCann, R. J. & Pass, B. (submitted, 2016) Multidimensional matching. Preprint at arXiv:1604.05771.
[2]Chiappori, P.-A., McCann, R. J. & Pass, B. (2017) Multi- to one-dimensional optimal transport. Commun. Pure Appl. Math. 70(12), 24052444.
[3]Gangbo, W. & McCann, R. J. (2000) Shape recognition via Wasserstein distance. Quart. Appl. Math. 58(4), 705737.
[4]McCann, R. J. & Pass, B. (submitted, 2018) Optimal transportation between unequal dimensions. Preprint at https://arxiv.org/abs/1805.11187.
[5]Pass, B. (2012) Regularity of optimal transportation between spaces with different dimensions. Math. Res. Lett. 19(2), 291307.
[6]Santambrogio, F. (2015) Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling. Progress in Nonlinear Differential Equations and their Applications, Vol. 87, Birkhäuser/Springer, Cham.
[7]Villani, C. (2003) Topics in Optimal Transportation. Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence.
[8]Villani, C. (2009) Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften, Vol. 338, Springer, New York.
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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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