A Variational Formulation of the BDF2 Method for Metric Gradient Flows

2017/11/08 by Matthes, Daniel, Plazotta, Simon · 3 citations
#35A15 (Primary) 34G25 #35G25 #35K46 #35Q84 #65J08 (Secondary) #65L06 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1711.02935

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 L²-Wasserstein metric.

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