| Title: |
Limits and consistency of non-local and graph approximations to the Eikonal equation |
| Authors: |
Fadili, Jalal M.; Forcadel, Nicolas; Tuyen Nguyen, Thi; Zantout, Rita |
| Contributors: |
Equipe Image - Laboratoire GREYC - UMR6072; Groupe de Recherche en Informatique, Image et Instrumentation de Caen (GREYC); Université de Caen Normandie (UNICAEN); Normandie Université (NU)-Normandie Université (NU)-École Nationale Supérieure d'Ingénieurs de Caen (ENSICAEN); Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN); Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS); Normandie Université (NU); École Nationale Supérieure d'Ingénieurs de Caen (ENSICAEN); Institut national des sciences appliquées Rouen Normandie (INSA Rouen Normandie); Institut National des Sciences Appliquées (INSA)-Normandie Université (NU); Laboratoire de Mathématiques de l'INSA de Rouen Normandie (LMI); Institut National des Sciences Appliquées (INSA)-Normandie Université (NU)-Institut National des Sciences Appliquées (INSA)-Normandie Université (NU); This work was supported by the Normandy Region grant MoNomads and partly by the European Union’sHorizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 777826 (NoMADS).; European Project: 777826,NoMADS(2018) |
| Source: |
ISSN: 0272-4979. |
| Publisher Information: |
CCSD; Oxford University Press (OUP) |
| Publication Year: |
2023 |
| Collection: |
Normandie Université: HAL |
| Subject Terms: |
Error bounds; Continuum limits; Weighted graphs; Viscosity solution; Non-local; Eikonal equation; MSC: 70H20; 49L25; 65N15; 58J32; 60D05; 05C90; [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]; [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing; [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA] |
| Description: |
International audience ; 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. |
| Document Type: |
article in journal/newspaper |
| Language: |
English |
| Relation: |
info:eu-repo/grantAgreement//777826/EU/Nonlocal Methods for Arbitrary Data Sources/NoMADS |
| DOI: |
10.1093/imanum/drac082 |
| Availability: |
https://hal.science/hal-03218100; https://hal.science/hal-03218100v5/document; https://hal.science/hal-03218100v5/file/Paper_finalversion_arxiv.pdf; https://doi.org/10.1093/imanum/drac082 |
| Rights: |
info:eu-repo/semantics/OpenAccess |
| Accession Number: |
edsbas.6DDBB669 |
| Database: |
BASE |