Mean curvature, threshold dynamics, and phase field theory on finite graphs
van Gennip, Yves and Guillen, Nestor and Osting, Braxton and Bertozzi, Andrea L. (2014) Mean curvature, threshold dynamics, and phase field theory on finite graphs. Milan Journal of Mathematics, 82 (1). pp. 3-65. ISSN 1424-9286
Official URL: http://dx.doi.org/10.1007/s00032-014-0216-8
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.
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