Nonlocal p-Laplacian evolution problems on graphs

Yosra Hafiene (1) , Jalal M. Fadili (2, 3, 1, 4) , Abderrahim Elmoataz (2)

1 ENSICAEN - École Nationale Supérieure d'Ingénieurs de Caen
2 Equipe Image - Laboratoire GREYC - UMR6072
3 NU - Normandie Université
4 GREYC - Groupe de Recherche en Informatique, Image et Instrumentation de Caen

In this paper we study numerical approximations of the evolution problem for the nonlocal p-Laplacian with homogeneous Neumann boundary conditions. First, we derive a bound on the distance between two continuous-in-time trajectories defined by two different evolution systems (i.e. with different kernels and initial data). We then provide a similar bound for the case when one of the trajectories is discrete-in-time and the other is continuous. In turn, these results allow us to establish error estimates of the discretized p-Laplacian problem on graphs. More precisely, for networks on convergent graph sequences (simple and weighted graphs), we prove convergence and provide rate of convergence of solutions for the discrete models to the solution of the continuous problem as the number of vertices grows. We finally touch on the limit as p → ∞ in these approximations and get uniform convergence results.

Mots clés: Nonlocal diffusion; p-Laplacian; graphs; graph limits; numerical approximation

Domaines: Traitement du signal et de l'image [eess.SP]; Théorie de l'information [cs.IT]; Théorie [stat.TH]; Machine Learning [stat.ML]; Analyse numérique [math.NA]; Analyse fonctionnelle [math.FA]; Statistiques [math.ST]; Optimisation et contrôle [math.OC]

Yosra Hafiene, Jalal M. Fadili, Abderrahim Elmoataz. Nonlocal p-Laplacian evolution problems on graphs. SIAM Journal on Numerical Analysis, 2018, 56 (2), pp.1064-1090. ⟨hal-01921849⟩