---
_id: '14703'
abstract:
- lang: eng
  text: We present a discretization of the dynamic optimal transport problem for which
    we can obtain the convergence rate for the value of the transport cost to its
    continuous value when the temporal and spatial stepsize vanish. This convergence
    result does not require any regularity assumption on the measures, though experiments
    suggest that the rate is not sharp. Via an analysis of the duality gap we also
    obtain the convergence rates for the gradient of the optimal potentials and the
    velocity field under mild regularity assumptions. To obtain such rates we discretize
    the dual formulation of the dynamic optimal transport problem and use the mature
    literature related to the error due to discretizing the Hamilton-Jacobi equation.
acknowledgement: "The authors would like to thank Chris Wojtan for his continuous
  support and several interesting discussions. Part of this research was performed
  during two visits: one of SI to the BIDSA research center at Bocconi University,
  and one of HL to the Institute of Science and Technology Austria. Both host institutions
  are warmly acknowledged for the hospital-\r\nity. HL is partially supported by the
  MUR-Prin 2022-202244A7YL “Gradient Flows and Non-Smooth Geometric Structures with
  Applications to Optimization and Machine Learning”, funded by the European Union
  - Next Generation EU. SI is supported in part by ERC Consolidator Grant 101045083
  “CoDiNA” funded by the European Research Council."
article_number: '2312.12213'
article_processing_charge: No
arxiv: 1
author:
- first_name: Sadashige
  full_name: Ishida, Sadashige
  id: 6F7C4B96-A8E9-11E9-A7CA-09ECE5697425
  last_name: Ishida
- first_name: Hugo
  full_name: Lavenant, Hugo
  last_name: Lavenant
citation:
  ama: Ishida S, Lavenant H. Quantitative convergence of a discretization of dynamic
    optimal transport using the dual formulation. <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/arXiv.2312.12213">10.48550/arXiv.2312.12213</a>
  apa: Ishida, S., &#38; Lavenant, H. (n.d.). Quantitative convergence of a discretization
    of dynamic optimal transport using the dual formulation. <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.2312.12213">https://doi.org/10.48550/arXiv.2312.12213</a>
  chicago: Ishida, Sadashige, and Hugo Lavenant. “Quantitative Convergence of a Discretization
    of Dynamic Optimal Transport Using the Dual Formulation.” <i>ArXiv</i>, n.d. <a
    href="https://doi.org/10.48550/arXiv.2312.12213">https://doi.org/10.48550/arXiv.2312.12213</a>.
  ieee: S. Ishida and H. Lavenant, “Quantitative convergence of a discretization of
    dynamic optimal transport using the dual formulation,” <i>arXiv</i>. .
  ista: Ishida S, Lavenant H. Quantitative convergence of a discretization of dynamic
    optimal transport using the dual formulation. arXiv, 2312.12213.
  mla: Ishida, Sadashige, and Hugo Lavenant. “Quantitative Convergence of a Discretization
    of Dynamic Optimal Transport Using the Dual Formulation.” <i>ArXiv</i>, 2312.12213,
    doi:<a href="https://doi.org/10.48550/arXiv.2312.12213">10.48550/arXiv.2312.12213</a>.
  short: S. Ishida, H. Lavenant, ArXiv (n.d.).
date_created: 2023-12-21T10:14:37Z
date_published: 2023-12-19T00:00:00Z
date_updated: 2023-12-27T13:44:33Z
day: '19'
department:
- _id: GradSch
- _id: ChWo
doi: 10.48550/arXiv.2312.12213
external_id:
  arxiv:
  - '2312.12213'
keyword:
- Optimal transport
- Hamilton-Jacobi equation
- convex optimization
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2312.12213
month: '12'
oa: 1
oa_version: Preprint
project:
- _id: 34bc2376-11ca-11ed-8bc3-9a3b3961a088
  grant_number: '101045083'
  name: Computational Discovery of Numerical Algorithms for Animation and Simulation
    of Natural Phenomena
publication: arXiv
publication_status: submitted
status: public
title: Quantitative convergence of a discretization of dynamic optimal transport using
  the dual formulation
type: preprint
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
year: '2023'
...