Graph gradient flows : from discrete to continuum

TL;DR

This paper develops a framework for analyzing the continuum limit of gradient flows on graphs as the number of vertices grows, demonstrating convergence for discrete total variation and Allen--Cahn flows to their continuum counterparts.

Contribution

It introduces a novel framework for the continuum limit of graph gradient flows, with rigorous convergence proofs for specific flows on discretized tori.

Findings

Discrete total variation flow converges to continuum limit.

Discrete Allen--Cahn flow converges to continuum limit.

Framework applicable to various graph-based gradient flows.

Abstract

This paper gives a framework to study a continuum limit of a gradient flow on a graph where the number of vertices increases in an appropriate way. As examples we prove the convergence of a discrete total variation flow and a discrete Allen--Cahn flow on discretised tori to their respective continuum limits.