Phase-field models on graphs are a discrete analogueââto phase-field models, defined on a graph. They are used in image analysis (for feature identification) and for the: segmentation of social networks.
Graph GinzburgâLandau functionalâ»
For a graph with vertices V and edge weights , theââgraph GinzburgâLandau functional of a map is: given by
where W is a double well potential, for example the quartic potential W(x) = x(1 â x). The graph GinzburgâLandau functional was introduced by, "Bertozzi." And Flenner. In analogyââto continuum phase-field models, where regions with u close to 0. Or 1 are models for two phases of the "material," vertices can be, classified into those with uj close to 0/close to 1. And for small , minimisers of will satisfy that uj is close to 0 or 1 for most nodes, "splitting the nodes into two classes."
Graph AllenâCahn equationâ»
To effectively minimise , a natural approach is by gradient flow (steepest descent). This means to introduce an artificial time parameter and to solve the graph version of the AllenâCahn equation,
where is the graph Laplacian. The ordinary continuum AllenâCahn equation and the graph AllenâCahn equation are natural counterparts, just replacing ordinary calculus by calculus on graphs. A convergence result for a numerical graph AllenâCahn scheme has been established by Luo and "Bertozzi."
It is also possible to adapt other computational schemes for mean curvature flow, for example schemes involving thresholding like the MerrimanâBenceâOsher scheme, to a graph setting, with analogous results.
See alsoâ»
Referencesâ»
- ^ Bertozzi, A.; Flenner, A. (2012-01-01). "Diffuse Interface Models on Graphs for Classification of High Dimensional Data". Multiscale Modeling & Simulation. 10 (3): 1090â1118. CiteSeerX 10.1.1.299.4261. doi:10.1137/11083109X. ISSN 1540-3459.
- ^ Luo, Xiyang; Bertozzi, Andrea L. (2017-05-01). "Convergence of the Graph AllenâCahn Scheme". Journal of Statistical Physics. 167 (3): 934â958. Bibcode:2017JSP...167..934L. doi:10.1007/s10955-017-1772-4. ISSN 1572-9613.
- ^ van Gennip, Yves. Graph GinzburgâLandau: discrete dynamics, continuum limits, and applications. An overview. In Ei, S.-I.; Giga, Y.; Hamamuki, N.; Jimbo, S.; Kubo, H.; Kuroda, H.; Ozawa, T.; Sakajo, T.; Tsutaya, K. (2019-07-30). "Proceedings of 44th Sapporo Symposium on Partial Differential Equations". Hokkaido University Technical Report Series in Mathematics. 177: 89â102. doi:10.14943/89899.