## 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 semisupervised learning Bertozzi and Flenner [Multiscale Model. Simul., 10 (2012), pp. 1090--1118] developed an algorithm based on the Allen--Cahn (AC) gradient flow of a graph Ginzburg--Landau functional, and Merkurjev, Kostić, and Bertozzi [SIAM J. Imaging Sci., 6 (2013), pp. 1903--1930] 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 AC flow. First, we choose the double-obstacle potential for the Ginzburg--Landau functional and derive well-posedness and regularity results for the resulting graph AC flow. Next, we exhibit a “semidiscrete” time-discretization scheme for AC flow of which the MBO scheme is a special case. We investigate the long-time behavior of this scheme and prove its convergence to the AC trajectory as the time-step vanishes. Finally, following a question raised by Van Gennip, Guillen, Osting, and Bertozzi [Milan J. Math., 82 (2014), pp. 3--65], we exhibit results toward proving a link between double-obstacle AC 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 [Proc. Math. Phys. Sci., 444 (1994), pp. 429--445] to prove the analogous link in the continuum.

Read More: https://epubs.siam.org/doi/10.1137/19M1277394

Read More: https://epubs.siam.org/doi/10.1137/19M1277394

Original language | English |
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Pages (from-to) | 4101–4139 |

Number of pages | 39 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 52 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2020 |

## Keywords

- Allen–Cahn equation
- Ginzburg–Landau functional
- Merriman–Bence–Osher scheme
- mean curvature flow
- double-obstacle potential
- graph dynamics
- Γ-convergence