Mass-conserving diffusion-based dynamics on graphs

An emerging technique in image segmentation, semi-supervised learning and general classification problems concerns the use of phase-separating flows defined on finite graphs. This was pioneered Bertozzi Flenner (2012, Multiscale Modeling Simulation 10 (3), 1090–1118), which used Allen–Cahn flow a graph, then extended Merkurjev et al. (2013, SIAM J. Imaging Sci. 6 (4), 1903–1930) using instead Merriman–Bence–Osher (MBO) scheme graph. In previous work by authors, Budd Van Gennip (2020, Math. Anal. 52 (5), 4101–4139), we gave theoretical justification for this MBO place flow, showing that is special case ‘semi-discrete’ numerical flow. paper, extend earlier work, link via semi-discrete robust to passing mass-conserving case. Inspired Rubinstein Sternberg (1992, IMA Appl. 48 , 249–264), define equation Then, with help tools convex optimisation, show our machinery can be applied derive graph as Allen–Cahn. We give analysis scheme, proving various desired properties like existence uniqueness convergence also yields choice function solutions scheme.