Volume 28, 2022
|Number of page(s)||49|
|Published online||07 July 2022|
A linear finite-difference scheme for approximating randers distances on cartesian grids*
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: firstname.lastname@example.org
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.
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