Guaranteeing Higher Order Convergence Rates for Accelerated Wasserstein Gradient Flow Schemes

TL;DR

This paper introduces a novel accelerated second-order scheme for Wasserstein gradient flows that achieves optimal quadratic convergence for smooth functionals and maintains strong stability under weaker assumptions.

Contribution

It presents the first rigorous proof of accelerated second-order convergence rates for Wasserstein gradient flows, extending stability and convergence results to less smooth functionals.

Findings

Achieves quadratic convergence for smooth energy functionals.

Maintains first-order convergence under weaker assumptions.

Energy decreases or nearly monotone along the iterates.

Abstract

In this paper, we study higher-order-accurate-in-time minimizing movements schemes for Wasserstein gradient flows. We introduce a novel accelerated second-order scheme, leveraging the differential structure of the Wasserstein space in both Eulerian and Lagrangian coordinates. For sufficiently smooth energy functionals, we show that our scheme provably achieves an optimal quadratic convergence rate. Under the weaker assumptions of Wasserstein differentiability and $\lambda$-displacement convexity (for any $\lambda\in \mathbb{R}$), we show that our scheme still achieves a first-order convergence rate and has strong numerical stability. In particular, we show that the energy is nearly monotone in general, while when the energy is $L$-smooth and $\lambda$-displacement convex (with $\lambda>0$), we prove the energy is non-increasing and the norm of the Wasserstein gradient is exponentially decreasing along the iterates. Taken together, our work provides the first fully rigorous proof of accelerated second-order convergence rates for smooth functionals and shows that the scheme performs no worse than the classical scheme JKO scheme for functionals that are $\lambda$-displacement convex and Wasserstein differentiable.