Print Email Facebook Twitter Mass-conserving diffusion-based dynamics on graphs Title Mass-conserving diffusion-based dynamics on graphs Author Budd, J.M. (TU Delft Mathematical Physics) van Gennip, Y. (TU Delft Mathematical Physics) Date 2021 Abstract 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 technique was pioneered in Bertozzi and Flenner (2012, Multiscale Modeling and Simulation 10(3), 1090-1118), which used the Allen-Cahn flow on a graph, and was then extended in Merkurjev et al. (2013, SIAM J. Imaging Sci. 6(4), 1903-1930) using instead the Merriman-Bence-Osher (MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip (2020, SIAM J. Math. Anal. 52(5), 4101-4139), we gave a theoretical justification for this use of the MBO scheme in place of Allen-Cahn flow, showing that the MBO scheme is a special case of a 'semi-discrete' numerical scheme for Allen-Cahn flow. In this paper, we extend this earlier work, showing that this link via the semi-discrete scheme is robust to passing to the mass-conserving case. Inspired by Rubinstein and Sternberg (1992, IMA J. Appl. Math. 48, 249-264), we define a mass-conserving Allen-Cahn equation on a graph. Then, with the help of the tools of convex optimisation, we show that our earlier machinery can be applied to derive the mass-conserving MBO scheme on a graph as a special case of a semi-discrete scheme for mass-conserving Allen-Cahn. We give a theoretical analysis of this flow and scheme, proving various desired properties like existence and uniqueness of the flow and convergence of the scheme, and also show that the semi-discrete scheme yields a choice function for solutions to the mass-conserving MBO scheme. Subject Allen-Cahn equationconvex optimisationgraph dynamicsmass constrained motionthreshold dynamics To reference this document use: http://resolver.tudelft.nl/uuid:442cd229-f4d8-466d-a094-3171d288b49f DOI https://doi.org/10.1017/S0956792521000061 ISSN 0956-7925 Source European Journal of Applied Mathematics, 33 (2022) (3), 423–471 Part of collection Institutional Repository Document type journal article Rights © 2021 J.M. Budd, Y. van Gennip Files PDF mass_conserving_diffusion ... graphs.pdf 656.94 KB Close viewer /islandora/object/uuid:442cd229-f4d8-466d-a094-3171d288b49f/datastream/OBJ/view