2012/04/23 by Yves van Gennip, Andrea L. Bertozzi, van Gennip, Yves +1 · 1 citation
Computer Science · Mathematics · #35Q56 #35R02 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Graph theory and applications #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1204.5220
openalex publication_date 2012/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We study Gamma-convergence of graph based Ginzburg-Landau functionals, both the limit for zero diffusive interface parameter epsilon->0 and the limit for infinite nodes in the graph m -> infinity. For general graphs we prove that in the limit epsilon -> 0 the graph cut objective function is recovered. We show that the continuum limit of this objective function on 4-regular graphs is related to the total variation seminorm and compare it with the limit of the discretized Ginzburg-Landau functional. For both functionals we also study the simultaneous limit epsilon -> 0 and m -> infinity, by expressing epsilon as a power of m and taking m -> infinity. Finally we investigate the continuum limit for a nonlocal means type functional on a completely connected graph.