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Asymptotic analysis of the Ginzburg-Landau functional on point clouds
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Theil, Florian and Thorpe, Matthew (2019) Asymptotic analysis of the Ginzburg-Landau functional on point clouds. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 149 (2). pp. 387-427. doi:10.1017/prm.2018.32
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Official URL: https://doi.org/10.1017/prm.2018.32
Abstract
The Ginzburg–Landau functional is a phase transition model which is suitable for classification type problems. We study the asymptotics of a sequence of Ginzburg–Landau functionals with anisotropic interaction potentials on point clouds Ψn where n denotes the number data points. In particular, we show the limiting problem, in the sense of Γ-convergence, is related to the total variation norm restricted to functions taking binary values, which can be understood as a surface energy. We generalize the result known for isotropic interaction potentials to the anisotropic case and add a result concerning the rate of convergence.
| Item Type: | Journal Article | ||||||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||
| Library of Congress Subject Headings (LCSH): | Functionals, Geometry, Analytic | ||||||||||||
| Journal or Publication Title: | Proceedings of the Royal Society of Edinburgh Section A: Mathematics | ||||||||||||
| Publisher: | Cambridge University Press | ||||||||||||
| ISSN: | 0308-2105 | ||||||||||||
| Official Date: | April 2019 | ||||||||||||
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| Volume: | 149 | ||||||||||||
| Number: | 2 | ||||||||||||
| Page Range: | pp. 387-427 | ||||||||||||
| DOI: | 10.1017/prm.2018.32 | ||||||||||||
| Status: | Peer Reviewed | ||||||||||||
| Publication Status: | Published | ||||||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
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