Limits and consistency of non-local and graph approximations to the Eikonal equation

Jalal M. Fadili (1, 2, 3, 4) , Nicolas Forcadel (2, 5, 6) , Thi Tuyen Nguyen (2, 5, 6) , Rita Zantout (2, 5, 6)

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 INSA Rouen Normandie - Institut national des sciences appliquées Rouen Normandie
6 LMI - Laboratoire de Mathématiques de l'INSA de Rouen Normandie

In this paper, we study a non-local approximation of the time-dependent (local) Eikonal equation with Dirichlet-type boundary conditions, where the kernel in the non-local problem is properly scaled. Based on the theory of viscosity solutions, we prove existence and uniqueness of the viscosity solutions of both the local and non-local problems, as well as regularity properties of these solutions in time and space. We then derive error bounds between the solution to the non-local problem and that of the local one, both in continuous-time and Backward Euler time discretization. We then turn to studying continuum limits of non-local problems defined on random weighted graphs with $n$ vertices. In particular, we establish that if the kernel scale parameter decreases at an appropriate rate as $n$ grows, then almost surely, the solution of the problem on graphs converges uniformly to the viscosity solution of the local problem as the time step vanishes and the number vertices $n$ grows large.

Mots clés: Eikonal equation; Non-local; Viscosity solution; Error bounds; Continuum limits; Weighted graphs

Domaines: Analyse numérique [math.NA]; Analyse fonctionnelle [math.FA]; Traitement du signal et de l'image [eess.SP]

Jalal M. Fadili, Nicolas Forcadel, Thi Tuyen Nguyen, Rita Zantout. Limits and consistency of non-local and graph approximations to the Eikonal equation. 2022. ⟨hal-03218100v3⟩