Discrete EVI and a bound between the JKO and entropic JKO schemes

par Sofiane Cherf (ICJ)

mercredi 21 mai 2025, 10:00 → 12:00 Europe/Paris

112 (Braconnier)

The Schrödinger problem (or entropic optimal transport problem) has been used to compute a regularized version of optimal transport, which can be computed very efficiently. To enable fast computations, one can attempt to replace the Wasserstein distance in the JKO scheme with its entropic counterpart. This modified scheme is called the entropic JKO scheme.

In a recent paper, Baradat, Hraivoronska, and Santambrogio proved the convergence of the entropic JKO scheme toward the limiting PDE of the classical JKO scheme, with an additional parabolic term, as the time discretization parameter tends to zero.

The goal of this talk is to explain how to obtain a bound in Wasserstein distance—in both the time discretization parameter and the regularization parameter—between the JKO scheme and the entropic JKO scheme under convexity assumptions. To this end, I will explore the contraction and stability properties of the JKO scheme.

Surprisingly, both the contraction and stability properties rely on the same inequality: the discrete Evolution Variational Inequality (EVI). Since this inequality plays a central role, I will dedicate a full part of the talk to it. I will try to explain where it comes from and present a classical application: the convergence rate of the JKO scheme toward its limiting PDE.