Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm

Jean-David Benamou (1, 2) , Guillaume Carlier (1, 2) , Luca Nenna (2, 3)

1 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 CNRS - Centre National de la Recherche Scientifique

Starting from Brenier's relaxed formulation of the incompress-ible Euler equation in terms of geodesics in the group of measure-preserving diffeomorphisms, we propose a numerical method based on Sinkhorn's algorithm for the entropic regularization of optimal transport. We also make a detailed comparison of this entropic regular-ization with the so-called Bredinger entropic interpolation problem (see [1]). Numerical results in dimension one and two illustrate the feasibility of the method.

Mots clés: Incompressible Euler equations; Generalized incompressible flows; Sinkhorn algorithm; Entropic regularization; Multi-marginal optimal transport

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Optimisation et contrôle [math.OC]

Jean-David Benamou, Guillaume Carlier, Luca Nenna. Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm. Numerische Mathematik, 2018. ⟨hal-01621311v2⟩