Continuum limit of the nonlocal <i>p</i>-Laplacian evolution problem on random inhomogeneous graphs

Hafiene, Yosra; Fadili, Jalal M.; Chesneau, Christophe; Elmoataz, Abderrahim

doi:10.1051/m2an/2019066

Cited by 9 publications

(12 citation statements)

References 24 publications

(59 reference statements)

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“…Studying consistency and continuum limits of certain evolution and variational problems on graphs and networks is an active research area; see [27,26,37,33,34,25,24,46,13] for a non-exhaustive list and references therein. In particular, the authors in [8,41] studied continumm limits of Lipschitz learning on graphs.…”

Section: Contributions and Relation To Prior Workmentioning

confidence: 99%

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

Fadili¹,

Forcadel²,

Nguyen³

et al. 2021

Preprint

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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 continuoustime 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.

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Section: Contributions and Relation To Prior Workmentioning

confidence: 99%

“…Since Γ ⊂ Ω, and in view of (H.7) and (H.9), we can choose K 2 ≥ C g c −1 g P L ∞ ( Ω\ Γ) + L ψ and η = aε (recall the definition of a from assumption (H.9)). Then continuing from (26), and using (H.9), we get…”

Section: γ Defmentioning

confidence: 99%

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

Fadili¹,

Forcadel²,

Nguyen³

et al. 2021

Preprint

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show abstract

“…They developed a theory of limits for sequences of sparse graphs based on such graphons, which generalizes both the existing theory of bounded graphons that are tailored to dense graph limits [30], and its extension in [9] to sparse graphs under a no dense spots assumptions. The latter graph model was studied in [25] in the context of continuum limits of p-Laplacian evolution problems on graphs. Nevertheless, the boundedness assumption of the graphon underlying these graph models is still highly restrictive.…”

Section: Q Graphons and Graph Limitsmentioning

confidence: 99%

“…Example 3.1. For an example that cannot be handled using L ∞ graphons, and thus does not enter in the framework of [24,25], consider a K-random graph model G(n, K, ρ n ) constructed according to Definition 3.1 with…”

Section: Sparse K-random Graph Modelsmentioning

confidence: 99%

Continuum limit of $p$-Laplacian evolution problems on graphs:$L^q$ graphons and sparse graphs

Bouchairi,

Fadili,

Elmoataz

2020

Preprint

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In this paper we study continuum limits of the discretized p-Laplacian evolution problem on sparse graphs with homogeneous Neumann boundary conditions. This extends the results of [24] to a far more general class of kernels, possibly singular, and graph sequences whose limit are the so-called L q -graphons. More precisely, we derive a bound on the distance between two continuous-in-time trajectories defined by two different evolution systems (i.e. with different kernels, second member and initial data). Similarly, we provide a bound in the case that one of the trajectories is discrete-in-time and the other is continuous. In turn, these results lead us to establish error estimates of the full discretization of the p-Laplacian problem on sparse random graphs. In particular, we provide rate of convergence of solutions for the discrete models to the solution of the continuous problem as the number of vertices grows.

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“…By virtue of [29,Lemma A.1], together with (A.2) and the fact that p ≥ 2, there exists a positive constant C 1 , such that for any β > 0…”

Section: Network On Graphs Generated By Deterministic Nodesmentioning

confidence: 99%

Continuum Limits of Nonlocal $p$-Laplacian Variational Problems on Graphs

Hafiene¹,

Fadili²,

Elmoataz³

2019

SIAM J. Imaging Sci.

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In this paper, we study a nonlocal variational problem which consists of minimizing in L 2 the sum of a quadratic data fidelity and a regularization term corresponding to the L p -norm of the nonlocal gradient. In particular, we study convergence of the numerical solution to a discrete version of this nonlocal variational problem to the unique solution of the continuum one. To do so, we derive an error bound and highlight the role of the initial data and the kernel governing the nonlocal interactions. When applied to variational problem on graphs, this error bound allows us to show the consistency of the discretized variational problem as the number of vertices goes to infinity. More precisely, for networks in convergent graph sequences (simple and weighted deterministic dense graphs as well as random inhomogeneous graphs), we prove convergence and provide rate of convergence of solutions for the discrete models to the solution of the continuum problem as the number of vertices grows.

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

Cited by 9 publications

References 24 publications

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

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

Continuum limit of $p$-Laplacian evolution problems on graphs:$L^q$ graphons and sparse graphs

Continuum Limits of Nonlocal $p$-Laplacian Variational Problems on Graphs

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