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Mean curvature, threshold dynamics, and phase field theory on finite graphs

van Gennip, Yves; Guillen, Nestor; Osting, Braxton; Bertozzi, Andrea L.

Authors

Yves van Gennip y.vangennip@nottingham.ac.uk

Nestor Guillen nestor@math.ucla.edu

Braxton Osting braxton@math.ucla.edu

Andrea L. Bertozzi bertozzi@math.ucla.edu



Abstract

In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study.

We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow.

We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the time-step and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as ``freezing'' or ``pinning'') and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize (continuum) nonlocal total variation.

Journal Article Type Article
Publication Date Jan 1, 2014
Journal Milan Journal of Mathematics
Print ISSN 1424-9286
Electronic ISSN 1424-9286
Publisher Humana Press
Peer Reviewed Peer Reviewed
Volume 82
Issue 1
APA6 Citation van Gennip, Y., Guillen, N., Osting, B., & Bertozzi, A. L. (2014). Mean curvature, threshold dynamics, and phase field theory on finite graphs. Milan Journal of Mathematics, 82(1), doi:10.1007/s00032-014-0216-8
DOI https://doi.org/10.1007/s00032-014-0216-8
Keywords spectral graph theory, Allen-Cahn equation, Ginzburg-Landau
functional, Merriman-Bence-Osher threshold dynamics, graph cut
function, total variation, mean curvature flow, nonlocal mean curvature, gamma convergence, graph coarea formula
Publisher URL http://dx.doi.org/10.1007/s00032-014-0216-8
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf
Additional Information The final publication is available at Springer via http://dx.doi.org/10.1007/s00032-014-0216-8

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





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