Nonlocal Continuum Limits of p-Laplacian Problems on Graphs

In this Element, the authors consider fully discretized p-Laplacian problems (evolution, boundary value and variational problems) on graphs.

Imad El Bouchairi (Author), Jalal Fadili (Author), Yosra Hafiene (Author), Abderrahim Elmoataz (Author)

9781009327855, Cambridge University Press

Paperback / softback, published 11 May 2023

75 pages
22.9 x 15.2 x 0.7 cm, 0.091 kg

In this Element, the authors consider fully discretized p-Laplacian problems (evolution, boundary value and variational problems) on graphs. The motivation of nonlocal continuum limits comes from the quest of understanding collective dynamics in large ensembles of interacting particles, which is a fundamental problem in nonlinear science, with applications ranging from biology to physics, chemistry and computer science. Using the theory of graphons, the authors give a unified treatment of all the above problems and establish the continuum limit for each of them together with non-asymptotic convergence rates. They also describe an algorithmic framework based proximal splitting to solve these discrete problems on graphs.

1. Introduction
2. Mathematical Background
3. Nonlocal p-Laplacian evolution problem on graphs
4. Nonlocal p-Laplacian variational problem on graphs
5. Nonlocal p-Laplacian Dirichlet problem on graphs
6. Algorithmic framework based on proximal splitting
References.

Subject Areas: Numerical analysis [PBKS]