Asymptotic analysis of the Ginzburg–Landau functional on point clouds. Issue 2 (27th December 2018)

• Record Type: Journal Article
• Authors: Thorpe, Matthew; Theil, Florian
• Is Part Of: Proceedings. Volume 149:Issue 2(2019)
• Page Start: 387
• Page End: 427
• Publication Date: 2018-12-27
• Subjects: Ginzburg-Landau Functional, -- Phase Transitions, -- PDEs on Graphs, -- Total Variation on Graphs, -- Large Data Asymptotics, -- Γ-Convergence; 49J45, -- 69J55, -- 62G20; Mathematics -- Periodicals; 510
• DOI: 10.1017/prm.2018.32
• Languages: English
• ISSNs: 0308-2105

Abstract

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 Ψ n 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.