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Mass-conserving diffusion-based dynamics on graphs

Part of: Graph theory Partial differential equations, boundary value problems Miscellaneous topics - Partial differential equations General theory in ordinary differential equations Boundary value problems

Published online by Cambridge University Press:  14 April 2021

J.M BUDD Open the ORCID record for J.M BUDD [Opens in a new window]  and
Y. VAN GENNIP Open the ORCID record for Y. VAN GENNIP [Opens in a new window]
J.M BUDD
Affiliation:
Delft Institute of Applied Mathematics (DIAM), Technische Universiteit Delft, Delft, The Netherlands e-mails: j.m.budd-1@tudelft.nl; y.vangennip@tudelft.nl
Y. VAN GENNIP
Affiliation:
Delft Institute of Applied Mathematics (DIAM), Technische Universiteit Delft, Delft, The Netherlands e-mails: j.m.budd-1@tudelft.nl; y.vangennip@tudelft.nl
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Abstract

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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 Simulation10(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.

Keywords

Allen–Cahn equationthreshold dynamicsgraph dynamicsmass constrained motionconvex optimisation

MSC classification

Primary: 34B45: Boundary value problems on graphs and networks 35R02: Partial differential equations on graphs and networks (ramified or polygonal spaces)
Secondary: 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions 65N12: Stability and convergence of numerical methods 05C99: None of the above, but in this section
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 3 , June 2022 , pp. 423 - 471
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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