A linear finite-difference scheme for approximating Randers distances on Cartesian grids.

Summary Using an extension of Varadhan's formula to Randers manifolds, we notice that Randers distances may be approximated by a logarithmic transformation of a linear second-order partial differential equation. Following an idea introduced by Crane, Weischedel, and Wardetzky in the case of Riemannian distances, we study a numerical method for approximating Randers distances which involves a discretization of this linear equation. We propose to use Selling's formula, which originates from the theory of low-dimensional lattice geometry, to build a monotone and linear finite-difference scheme. By injecting the logarithmic transformation in this linear scheme, we are able to prove convergence of this numerical method to the Randers distance, as well as consistency to the order two thirds far from the boundary of the considered domain. We explain how this method may be used to approximate optimal transport distances, how has been previously done in the Riemannian case.