Almost Sharp Rates of Convergence for the Average Cost and Displacement in the Optimal Matching Problem

Michael Goldman; Martin Huesmann; Felix Otto

doi:10.1007/978-3-031-91282-5_5

Almost Sharp Rates of Convergence for the Average Cost and Displacement in the Optimal Matching Problem

Michael Goldman
, Martin Huesmann
, Felix Otto

University of Münster
Max Planck Institute for Mathematics in the Sciences

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

In this note we prove estimates for the average cost in the quadratic optimal transport problem on the two-dimensional flat torus which are optimal up to a double logarithm. We also prove sharp estimates on the displacement. This is based on the combination of a post-processing of our quantitative linearization result together with a quasi-orthogonality property.

Original language	English
Title of host publication	Partial Differential Equations
Subtitle of host publication	Waves, Nonlinearities and Nonlocalities - The Abel Symposium 2023
Editors	Mats Ehrnström, Helge Holden, Espen R. Jakobsen
Publisher	Springer Science and Business Media Deutschland GmbH
Pages	93-109
Number of pages	17
ISBN (Print)	9783031912818
DOIs	https://doi.org/10.1007/978-3-031-91282-5_5
Publication status	Published - 1 Jan 2025
Event	Abel Symposium on Partial Differential Equations: Waves, Nonlinearities and Nonlocalities, 2023 - Orkanger, Norway Duration: 13 Jun 2023 → 16 Jun 2023

Publication series

Name	Abel Symposia
Volume	18 ABEL
ISSN (Print)	2193-2808
ISSN (Electronic)	2197-8549

Conference

Conference	Abel Symposium on Partial Differential Equations: Waves, Nonlinearities and Nonlocalities, 2023
Country/Territory	Norway
City	Orkanger
Period	13/06/23 → 16/06/23

Access to Document

10.1007/978-3-031-91282-5_5

Cite this

Goldman, M., Huesmann, M., & Otto, F. (2025). Almost Sharp Rates of Convergence for the Average Cost and Displacement in the Optimal Matching Problem. In M. Ehrnström, H. Holden, & E. R. Jakobsen (Eds.), Partial Differential Equations: Waves, Nonlinearities and Nonlocalities - The Abel Symposium 2023 (pp. 93-109). (Abel Symposia; Vol. 18 ABEL). Springer Science and Business Media Deutschland GmbH. https://doi.org/10.1007/978-3-031-91282-5_5

Goldman, Michael ; Huesmann, Martin ; Otto, Felix. / Almost Sharp Rates of Convergence for the Average Cost and Displacement in the Optimal Matching Problem. Partial Differential Equations: Waves, Nonlinearities and Nonlocalities - The Abel Symposium 2023. editor / Mats Ehrnström ; Helge Holden ; Espen R. Jakobsen. Springer Science and Business Media Deutschland GmbH, 2025. pp. 93-109 (Abel Symposia).

@inproceedings{e2c46cca5ded4a6990d1de6490685b8f,

title = "Almost Sharp Rates of Convergence for the Average Cost and Displacement in the Optimal Matching Problem",

abstract = "In this note we prove estimates for the average cost in the quadratic optimal transport problem on the two-dimensional flat torus which are optimal up to a double logarithm. We also prove sharp estimates on the displacement. This is based on the combination of a post-processing of our quantitative linearization result together with a quasi-orthogonality property.",

author = "Michael Goldman and Martin Huesmann and Felix Otto",

note = "Publisher Copyright: {\textcopyright} The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.; Abel Symposium on Partial Differential Equations: Waves, Nonlinearities and Nonlocalities, 2023 ; Conference date: 13-06-2023 Through 16-06-2023",

year = "2025",

month = jan,

day = "1",

doi = "10.1007/978-3-031-91282-5\_5",

language = "English",

isbn = "9783031912818",

series = "Abel Symposia",

publisher = "Springer Science and Business Media Deutschland GmbH",

pages = "93--109",

editor = "Mats Ehrnstr{\"o}m and Helge Holden and Jakobsen, \{Espen R.\}",

booktitle = "Partial Differential Equations",

}

Goldman, M, Huesmann, M & Otto, F 2025, Almost Sharp Rates of Convergence for the Average Cost and Displacement in the Optimal Matching Problem. in M Ehrnström, H Holden & ER Jakobsen (eds), Partial Differential Equations: Waves, Nonlinearities and Nonlocalities - The Abel Symposium 2023. Abel Symposia, vol. 18 ABEL, Springer Science and Business Media Deutschland GmbH, pp. 93-109, Abel Symposium on Partial Differential Equations: Waves, Nonlinearities and Nonlocalities, 2023, Orkanger, Norway, 13/06/23. https://doi.org/10.1007/978-3-031-91282-5_5

Almost Sharp Rates of Convergence for the Average Cost and Displacement in the Optimal Matching Problem. / Goldman, Michael; Huesmann, Martin; Otto, Felix.
Partial Differential Equations: Waves, Nonlinearities and Nonlocalities - The Abel Symposium 2023. ed. / Mats Ehrnström; Helge Holden; Espen R. Jakobsen. Springer Science and Business Media Deutschland GmbH, 2025. p. 93-109 (Abel Symposia; Vol. 18 ABEL).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

T1 - Almost Sharp Rates of Convergence for the Average Cost and Displacement in the Optimal Matching Problem

AU - Goldman, Michael

AU - Huesmann, Martin

AU - Otto, Felix

N1 - Publisher Copyright: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.

PY - 2025/1/1

Y1 - 2025/1/1

N2 - In this note we prove estimates for the average cost in the quadratic optimal transport problem on the two-dimensional flat torus which are optimal up to a double logarithm. We also prove sharp estimates on the displacement. This is based on the combination of a post-processing of our quantitative linearization result together with a quasi-orthogonality property.

AB - In this note we prove estimates for the average cost in the quadratic optimal transport problem on the two-dimensional flat torus which are optimal up to a double logarithm. We also prove sharp estimates on the displacement. This is based on the combination of a post-processing of our quantitative linearization result together with a quasi-orthogonality property.

UR - https://www.scopus.com/pages/publications/105014495038

U2 - 10.1007/978-3-031-91282-5_5

DO - 10.1007/978-3-031-91282-5_5

M3 - Conference contribution

AN - SCOPUS:105014495038

SN - 9783031912818

T3 - Abel Symposia

SP - 93

EP - 109

BT - Partial Differential Equations

A2 - Ehrnström, Mats

A2 - Holden, Helge

A2 - Jakobsen, Espen R.

PB - Springer Science and Business Media Deutschland GmbH

T2 - Abel Symposium on Partial Differential Equations: Waves, Nonlinearities and Nonlocalities, 2023

Y2 - 13 June 2023 through 16 June 2023

ER -

Goldman M, Huesmann M, Otto F. Almost Sharp Rates of Convergence for the Average Cost and Displacement in the Optimal Matching Problem. In Ehrnström M, Holden H, Jakobsen ER, editors, Partial Differential Equations: Waves, Nonlinearities and Nonlocalities - The Abel Symposium 2023. Springer Science and Business Media Deutschland GmbH. 2025. p. 93-109. (Abel Symposia). doi: 10.1007/978-3-031-91282-5_5