GRAPH MERRIMAN-BENCE-OSHER AS A SEMIDISCRETE IMPLICIT EULER SCHEME FOR GRAPH ALLEN-CAHN FLOW.

In recent years there has been an emerging interest in PDE-like ows 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 ow of a graph Ginzburg{Landau functional, and Merkurjev, Kostic, 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 ow. First, we choose the double-obstacle potential for the Ginzburg{Landau functional and derive wellposedness and regularity results for the resulting graph AC ow. Next, we exhibit a \semidiscrete" time-discretization scheme for AC ow 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 ow and mean curvature ow on graphs. We show some promising -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. [ABSTRACT FROM AUTHOR]