The first eigenvalue and eigenfunction of a nonlinear elliptic system

Bozorgnia, Farid; Mohammadi, Seyyed Abbas; Vejchodský, Tomáš

doi:10.1016/j.apnum.2019.06.004

Cited by 7 publications

(3 citation statements)

References 21 publications

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“…The matrix Δ (k) p is SPD (see [4,37]) which has its own eigendecomposition, offering a graph spectral convolution interpretation. Remark 3 Eq.…”

Section: Lemmamentioning

confidence: 99%

Enhancing framelet GCNs with generalized p-Laplacian regularization

Shao,

Shi,

Han

et al. 2023

Int. J. Mach. Learn. & Cyber.

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Graph neural networks (GNNs) have achieved remarkable results for various graph learning tasks. However, one of the recent challenges for GNNs is to adapt to different types of graph inputs, such as heterophilic graph datasets in which linked nodes are more likely to contain a different class of labels and features. Accordingly, an ideal GNN model should adaptively accommodate all types of graph datasets with different labeling distributions. In this paper, we tackle this challenge by proposing a regularization framework on graph framelet with the regularizer induced from graph p-Laplacian. By adjusting the value of p, the p-Laplacian based regularizer restricts the solution space of graph framelet into the desirable region based on the graph homophilic features. We propose an algorithm to effectively solve a more generalized regularization problem and prove that the algorithm imposes a (p-Laplacian based) spectral convolution and diagonal scaling operation to the framelet filtered node features. Furthermore, we analyze the denoising power of the proposed model and compare it with the predefined framelet denoising regularizer. Finally, we conduct empirical studies to show the prediction power of the proposed model in both homophily undirect and heterophily direct graphs with and without noises. Our proposed model shows significant improvements compared to multiple baselines, and this suggests the effectiveness of combining graph framelet and p-Laplacian.

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“…The matrix Δ (k) p is SPD (see [4,37]) which has its own eigendecomposition, offering a graph spectral convolution interpretation. Remark 3 Eq.…”

Section: Lemmamentioning

confidence: 99%

Enhancing framelet GCNs with generalized p-Laplacian regularization

Shao,

Shi,

Han

et al. 2023

Int. J. Mach. Learn. & Cyber.

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show abstract

“…The inverse power method is known to be an efficient method to approximate the first eigenvalue of given operator see [8,9,18]. Our aim here is to extend this method combining with Lemma 2.1 to approximate the second eigenvalue under symmetry assumption on domain.…”

Section: An Iterative Schemementioning

confidence: 99%

Approximation of the second eigenvalue of the $p$-Laplace operator in symmetric domains

Bozorgnia¹

2019

Preprint

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“…Now, according to a recent result by Bozorgnia et al [6,Theorem 3.3] obtained in the case of constant exponents, the following situation may occur min λ 1,p 1 , λ 1,p 2 < λ 1 (p). Therefore, in the case of constant exponents, condition (1.3) is weaker than condition (1.1).…”

Section: Introductionmentioning

confidence: 96%

Solvability of Parametric Elliptic Systems with Variable Exponents

Ouannasser,

El Hachimi

2023

Moroccan Journal of Pure and Applied Analysis

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In this paper, we study the solvability to the left of the positive infimum of all eigenvalues for some non-resonant quasilinear elliptic problems involving variable exponents. We first prove the existence of at least a weak solution for some non-variational systems by using a surjectivity result for pseudomonotone operators. Furthermore, under additional conditions, we show that the solution is unique and provide examples. Second, we deal with non-resonant gradient-type systems and obtain existence by using a variational approach.

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The first eigenvalue and eigenfunction of a nonlinear elliptic system

Cited by 7 publications

References 21 publications

Enhancing framelet GCNs with generalized p-Laplacian regularization

Enhancing framelet GCNs with generalized p-Laplacian regularization

Approximation of the second eigenvalue of the $p$-Laplace operator in symmetric domains

Solvability of Parametric Elliptic Systems with Variable Exponents

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