# Graph MBO as a semi-discrete implicit Euler scheme for graph Allen-Cahn flow

## TL;DR

This paper rigorously justifies the use of the MBO scheme as a semi-discrete implicit Euler method for graph Allen-Cahn flow, establishing well-posedness, convergence, and links to mean curvature flow on graphs.

## Contribution

It provides a rigorous mathematical foundation for the MBO scheme as a time discretization of graph Allen-Cahn flow, including well-posedness, convergence, and geometric flow connections.

## Key findings

- MBO scheme is a special case of a semi-discrete scheme for Allen-Cahn flow.
- Proved convergence of the scheme to the Allen-Cahn trajectory as time-step decreases.
- Established links between double-obstacle Allen-Cahn flow and graph mean curvature flow.

## Abstract

In recent years there has been an emerging interest in PDE-like flows defined on finite graphs, with applications in clustering and image segmentation. In particular for image segmentation and semi-supervised learning Bertozzi and Flenner (2012) developed an algorithm based on the Allen-Cahn gradient flow of a graph Ginzburg-Landau functional, and Merkurjev, Kosti\'c and Bertozzi (2013) devised a variant algorithm based instead on graph Merriman-Bence-Osher (MBO) dynamics. This work offers rigorous justification for this use of the MBO scheme in place of Allen-Cahn flow. First, we choose the double-obstacle potential for the Ginzburg-Landau functional, and derive well-posedness and regularity results for the resulting graph Allen-Cahn flow. Next, we exhibit a "semi-discrete" time-discretisation scheme for Allen-Cahn flow of which the MBO scheme is a special case. We investigate the long-time behaviour of this scheme, and prove its convergence to the Allen-Cahn trajectory as the time-step vanishes. Finally, following a question raised by Van Gennip, Guillen, Osting and Bertozzi (2014), we exhibit results towards proving a link between double-obstacle Allen-Cahn flow and mean curvature flow on graphs. We show some promising $\Gamma$-convergence results, and translate to the graph setting two comparison principles used by Chen and Elliott (1994) to prove the analogous link in the continuum.