On the total variation Wasserstein gradient flow and the TV-JKO scheme

Guillaume Carlier (1, 2) , Clarice Poon (1, 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 DAMTP - Department of Applied Mathematics and Theoretical Physics

We study the JKO scheme for the total variation, characterize the optimizers, prove some of their qualitative properties (in particular a sort of maximum principle and the regularity of level sets). We study in detail the case of step functions. Finally, in dimension one, we establish convergence as the time step goes to zero to a solution of a fourth-order nonlinear evolution equation.

Mots clés: Fourth-order evolution equations; JKO scheme; Total variation; Wasserstein gradient flows

Domaines: Optimisation et contrôle [math.OC]

Guillaume Carlier, Clarice Poon. On the total variation Wasserstein gradient flow and the TV-JKO scheme. ESAIM: Control, Optimisation and Calculus of Variations, 2019. ⟨hal-01492343⟩