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Asymptotic analysis of the Ginzburg–Landau functional on point clouds

Part of: Nonparametric inference Existence theories

Published online by Cambridge University Press:  27 December 2018

Matthew Thorpe  and
Florian Theil
Matthew Thorpe
Affiliation:
University of Warwick, Coventry, CV4 7AL, UK ([email protected])
Florian Theil
Affiliation:
University of Warwick, Coventry, CV4 7AL, UK ([email protected])
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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.

Keywords

Ginzburg-Landau FunctionalPhase TransitionsPDEs on GraphsTotal Variation on GraphsLarge Data AsymptoticsΓ-Convergence

MSC classification

Primary: 49J45: Methods involving semicontinuity and convergence; relaxation
Secondary: 62G20: Asymptotic properties
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 2 , April 2019 , pp. 387 - 427
Copyright
Copyright © Royal Society of Edinburgh 2018 

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Footnotes

*

Present address: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, email: [email protected]

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