2018 # 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 …

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2023

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“…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]).…”

confidence: 99%

“…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]).…”

confidence: 99%

“…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).…”

confidence: 99%

“…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…”

confidence: 99%

“…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.…”

confidence: 99%