Continuum limit of the nonlocal p-Laplacian evolution problem on random inhomogeneous graphs

Hafiene Yosra (1) , Jalal M. Fadili (1, 2, 3, 4) , Christophe Chesneau (5, 2, 6) , Abderrahim Elmoataz (1)

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

In this paper we study numerical approximations of the evolution problem for the nonlocal p-Laplacian operator with homogeneous Neumann boundary conditions on inhomogeneous random conver-gent graph sequences. More precisely, for networks on convergent inhomogeneous random graph sequences (generated first by deterministic and then random node sequences), we establish their continuum limits and provide rate of convergence of solutions for the discrete models to their continuum counterparts as the number of vertices grows. Our bounds reveals the role of the different parameters, and in particular that of p and the geometry/regularity of the data.

Mots clés: Nonlocal diffusion; p-Laplacian; inhomogeneous random 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]

Hafiene Yosra, Jalal M. Fadili, Christophe Chesneau, Abderrahim Elmoataz. Continuum limit of the nonlocal p-Laplacian evolution problem on random inhomogeneous graphs. ESAIM: Mathematical Modelling and Numerical Analysis, In press. ⟨hal-02276108⟩