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Journal Articles ESAIM: Control, Optimisation and Calculus of Variations Year : 2022

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

Abstract

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 two and three, which is consistent at order two thirds, 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.
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Dates and versions

hal-03125879 , version 1 (29-01-2021)
hal-03125879 , version 2 (09-06-2021)
hal-03125879 , version 3 (21-03-2022)

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Frédéric Bonnans, Guillaume Bonnet, Jean-Marie Mirebeau. A linear finite-difference scheme for approximating Randers distances on Cartesian grids. ESAIM: Control, Optimisation and Calculus of Variations, 2022, 28, ⟨10.1051/cocv/2022043⟩. ⟨hal-03125879v3⟩
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