A Variational Formulation of the BDF2 Method for Metric Gradient Flows

TL;DR

This paper introduces a variational formulation of the BDF2 method for gradient flows in metric spaces, proving convergence under weak assumptions and demonstrating its effectiveness through numerical experiments.

Contribution

It presents a novel variational BDF2 scheme for metric gradient flows, establishing well-posedness and convergence with a convergence order of one-half.

Findings

Proven convergence of the method in general metric spaces.

Numerical validation on Riemannian manifolds and Wasserstein spaces.

Demonstrated robustness without requiring smoothness of the energy functional.

Abstract

We propose a variational form of the BDF2 method as an alternative to the commonly used minimizing movement scheme for the time-discrete approximation of gradient flows in abstract metric spaces. Assuming uniform semi-convexity --- but no smoothness --- of the augmented energy functional, we prove well-posedness of the method and convergence of the discrete approximations to a curve of steepest descent. In a smooth Hilbertian setting, classical theory would predict a convergence order of two in time, we prove convergence order of one-half in the general metric setting and under our weak hypotheses. Further, we illustrate these results with numerical experiments for gradient flows on a compact Riemannian manifold, in a Hilbert space, and in the $L^2$-Wasserstein metric.