Nonlocal $p$-Laplacian Evolution Problems on Graphs

Abstract

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.

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

“…Throughout the paper, we will assume that ∈]1, +∞[. Existence and uniqueness of a strong solution to ( ) in the space (Ω) was shown in Theorem 3.1 of [17] (relying on arguments from [2]).…”

“…In [17], we provided a rigorous justification of the continuum limit ( ) for the discrete -Laplacian on deterministic dense graphs (graphs with vertices and Θ( 2 ) edges 6 ). The analysis of the continuum limit in [17] uses ideas from the theory of dense graph limits [6,21,22], which for every convergent family of dense graphs defines the limiting object, a measurable symmetric bounded and nonnegative function called graphon (see Sect. 2 for a brief overview on graphons).…”

“…This object characterizes the completion of the space of all graphs with respect to an appropriate metric. In [17], for convergent sequences of deterministic dense graphs { } ∈N , it was shown that with the kernel in ( ) taken to be the graphon associated to { } ∈N * , the solution of ( ) is well-approximated by those of the totally discrete problems ( , ). We gave precise convergence rates as a function of the discretization time step .…”

Partial differential equations and variational methods for geometric processing of images

“…The techniques developed to associate point cloud based functions with continuum functions have recently also been applied to prove consistency of other statistical methods [40,41] and to show that certain artificial neural networks have continuum limits that take the form of ODEconstrained variational models [79]. For discrete-to-continuum limit results, also graphon methods have been considered [45,46,58] 1.6. Further applications…”

Evolution problems of Leray–Lions type with nonhomogeneous Neumann boundary conditions in metric random walk spaces

“…Nonlocal diffusion problems of p-Laplacian type with homogeneous Neumann boundary conditions have been studied, see Examples 1.1 and 1.2 for the notation, in [R N , d, m J ] (see, for example, [4], [5]) and in graphs [V (G), d G , (m G x )] (see, for example, the work of Hafiene, Fadili and Elmoataz [16]) with the formulation u t (t, x) = Ω |u(y) − u(x)| p−2 (u(y) − u(x))dm x (y), x ∈ Ω, 0 < t < T.…”