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large data asymptotics

Proceedings of the Royal Society of Edinburgh Section A: Mathematics (1)

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1 results

Asymptotic analysis of the Ginzburg–Landau functional on point clouds
Part of
- Existence theories
- Nonparametric inference
Matthew Thorpe, Florian Theil
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 149 / Issue 2 / April 2019

Published online by Cambridge University Press:

27 December 2018, pp. 387-427

Print publication:

April 2019
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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.