| Issue |
ESAIM: COCV
Volume 28, 2022
|
|
|---|---|---|
| Article Number | 45 | |
| Number of page(s) | 49 | |
| DOI | https://doi.org/10.1051/cocv/2022043 | |
| Published online | 07 July 2022 | |
A linear finite-difference scheme for approximating randers distances on cartesian grids*
1
Inria-Saclay and CMAP, École Polytechnique, Palaiseau, France
2
LMO, Université Paris-Saclay, Orsay, France
3
Université Paris-Saclay, ENS Paris-Saclay, CNRS, Centre Borelli, Gif-sur-Yvette, France
** Corresponding author: guillaume.bonnet1@universite-paris-saclay.fr
Received:
10
June
2021
Accepted:
31
May
2022
Randers distances are an asymmetric generalization of Riemannian distances, and arise in optimal control problems subject to a drift term, among other applications. We show that Randers eikonal equation can be approximated by a logarithmic transformation of an anisotropic second order linear equation, generalizing Varadhan’s formula for Riemannian manifolds. Based on this observation, we establish the convergence of a numerical method for computing Randers distances, from point sources or from a domain’s boundary, on Cartesian grids of dimension 2 and 3, which is consistent at order 2/3, and uses tools from low-dimensional algorithmic geometry for best efficiency. We also propose a numerical method for optimal transport problems whose cost is a Randers distance, exploiting the linear structure of our discretization and generalizing previous works in the Riemannian case. Numerical experiments illustrate our results.
Mathematics Subject Classification: 65N06 / 65N12 / 49L25
Key words: Randers metric / Varadhan’s formula / Hamilton-Jacobi equation / viscosity solutions / finite-difference scheme / convergence analysis
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.