---
_id: '7573'
abstract:
- lang: eng
  text: This paper deals with dynamical optimal transport metrics defined by spatial
    discretisation of the Benamou–Benamou formula for the Kantorovich metric . Such
    metrics appear naturally in discretisations of -gradient flow formulations for
    dissipative PDE. However, it has recently been shown that these metrics do not
    in general converge to , 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.
acknowledgement: J.M. gratefully acknowledges support by the European Research Council
  (ERC) under the European Union's Horizon 2020 research and innovation programme
  (grant agreement No 716117). J.M. and L.P. also acknowledge support from the Austrian
  Science Fund (FWF), grants No F65 and W1245. E.K. gratefully acknowledges support
  by the German Research Foundation through the Hausdorff Center for Mathematics and
  the Collaborative Research Center 1060. P.G. is partially funded by the Deutsche
  Forschungsgemeinschaft (DFG, German Research Foundation) – 350398276.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Peter
  full_name: Gladbach, Peter
  last_name: Gladbach
- first_name: Eva
  full_name: Kopfer, Eva
  last_name: Kopfer
- first_name: Jan
  full_name: Maas, Jan
  id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
  last_name: Maas
  orcid: 0000-0002-0845-1338
- first_name: Lorenzo
  full_name: Portinale, Lorenzo
  id: 30AD2CBC-F248-11E8-B48F-1D18A9856A87
  last_name: Portinale
citation:
  ama: Gladbach P, Kopfer E, Maas J, Portinale L. Homogenisation of one-dimensional
    discrete optimal transport. <i>Journal de Mathematiques Pures et Appliquees</i>.
    2020;139(7):204-234. doi:<a href="https://doi.org/10.1016/j.matpur.2020.02.008">10.1016/j.matpur.2020.02.008</a>
  apa: Gladbach, P., Kopfer, E., Maas, J., &#38; Portinale, L. (2020). Homogenisation
    of one-dimensional discrete optimal transport. <i>Journal de Mathematiques Pures
    et Appliquees</i>. Elsevier. <a href="https://doi.org/10.1016/j.matpur.2020.02.008">https://doi.org/10.1016/j.matpur.2020.02.008</a>
  chicago: Gladbach, Peter, Eva Kopfer, Jan Maas, and Lorenzo Portinale. “Homogenisation
    of One-Dimensional Discrete Optimal Transport.” <i>Journal de Mathematiques Pures
    et Appliquees</i>. Elsevier, 2020. <a href="https://doi.org/10.1016/j.matpur.2020.02.008">https://doi.org/10.1016/j.matpur.2020.02.008</a>.
  ieee: P. Gladbach, E. Kopfer, J. Maas, and L. Portinale, “Homogenisation of one-dimensional
    discrete optimal transport,” <i>Journal de Mathematiques Pures et Appliquees</i>,
    vol. 139, no. 7. Elsevier, pp. 204–234, 2020.
  ista: Gladbach P, Kopfer E, Maas J, Portinale L. 2020. Homogenisation of one-dimensional
    discrete optimal transport. Journal de Mathematiques Pures et Appliquees. 139(7),
    204–234.
  mla: Gladbach, Peter, et al. “Homogenisation of One-Dimensional Discrete Optimal
    Transport.” <i>Journal de Mathematiques Pures et Appliquees</i>, vol. 139, no.
    7, Elsevier, 2020, pp. 204–34, doi:<a href="https://doi.org/10.1016/j.matpur.2020.02.008">10.1016/j.matpur.2020.02.008</a>.
  short: P. Gladbach, E. Kopfer, J. Maas, L. Portinale, Journal de Mathematiques Pures
    et Appliquees 139 (2020) 204–234.
date_created: 2020-03-08T23:00:47Z
date_published: 2020-07-01T00:00:00Z
date_updated: 2026-04-08T07:00:03Z
day: '01'
department:
- _id: JaMa
doi: 10.1016/j.matpur.2020.02.008
ec_funded: 1
external_id:
  arxiv:
  - '1905.05757'
  isi:
  - '000539439400008'
fulldoi: https://doi.org/10.1016/j.matpur.2020.02.008
intvolume: '       139'
isi: 1
issue: '7'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1905.05757
month: '07'
oa: 1
oa_version: Preprint
page: 204-234
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '716117'
  name: Optimal Transport and Stochastic Dynamics
- _id: 260482E2-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: F06504
  name: Taming Complexity in Partial Differential Systems
- _id: 260788DE-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: W1245
  name: Dissipation and dispersion in nonlinear partial differential equations
publication: Journal de Mathematiques Pures et Appliquees
publication_identifier:
  issn:
  - 0021-7824
publication_status: published
publisher: Elsevier
quality_controlled: '1'
related_material:
  record:
  - id: '10030'
    relation: dissertation_contains
    status: public
scopus_import: '1'
status: public
title: Homogenisation of one-dimensional discrete optimal transport
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 139
year: '2020'
...