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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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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03125879
Contributor : Guillaume Bonnet Connect in order to contact the contributor
Submitted on : Monday, March 21, 2022 - 2:09:23 PM
Last modification on : Tuesday, April 19, 2022 - 3:23:54 PM

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  • HAL Id : hal-03125879, version 3

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Frédéric Bonnans, Guillaume Bonnet, Jean-Marie Mirebeau. A linear finite-difference scheme for approximating Randers distances on Cartesian grids. 2022. ⟨hal-03125879v3⟩

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