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Journal Articles ESAIM: Mathematical Modelling and Numerical Analysis Year : 2022

Computing the cut locus of a Riemannian manifold via optimal transport

Abstract

In this paper, we give a new characterization of the cut locus of a point on a compact Riemannian manifold as the zero set of the optimal transport density solution of the Monge–Kantorovich equations, a PDE formulation of the optimal transport problem with cost equal to the geodesic distance. Combining this result with an optimal transport numerical solver, based on the so-called dynamical Monge–Kantorovich approach, we propose a novel framework for the numerical approximation of the cut locus of a point in a manifold. We show the applicability of the proposed method on a few examples settled on 2d-surfaces embedded in ℝ3, and discuss advantages and limitations.
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Dates and versions

hal-03778833 , version 1 (06-12-2021)
hal-03778833 , version 2 (16-09-2022)

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Enrico Facca, Luca Berti, Francesco Fassò, Mario Putti. Computing the cut locus of a Riemannian manifold via optimal transport. ESAIM: Mathematical Modelling and Numerical Analysis, 2022, 56 (6), pp.1939-1954. ⟨10.1051/m2an/2022059⟩. ⟨hal-03778833v2⟩
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