A numerical solution to Monge's problem with a Finsler distance as cost

Jean-David Benamou (1, 2) , Guillaume Carlier (1) , Roméo Hatchi (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales

Monge's problem with a Finsler cost is intimately related to an optimal flow problem. Discretization of this problem and its dual leads to a well-posed finite-dimensional saddle-point problem which can be solved numerically relatively easily by an augmented Lagrangian approach in the same spirit as the Benamou-Brenier method for the optimal transport problem with quadratic cost. Numerical results validate the method. We also emphasize that the algorithm only requires elementary operations and in particular never involves evaluation of the Finsler distance or of geodesics.

Mots clés: Monge's problem; Finsler distance; augmented Lagrangian MS Classification: 65K10; 90C25; 90C46

Domaines: Optimisation et contrôle [math.OC]

Jean-David Benamou, Guillaume Carlier, Roméo Hatchi. A numerical solution to Monge's problem with a Finsler distance as cost. ESAIM: Mathematical Modelling and Numerical Analysis, 2018. ⟨hal-01261094⟩