Approximation theorem for principle eigenvalue of discrete p-Laplacian

Abstract

For the principle eigenvalue of discrete weighted p-Laplacian on the set of nonnegative integers, the convergence of an approximation procedure and the inverse iteration is proved. Meanwhile, in the proof of the convergence, the monotonicity of an approximation sequence is also checked. To illustrate these results, some examples are presented.

Keywords

Principle eigenvalue / weighted p-Laplacian / inverse iteration / approximation theorem

References

[1] Biezuner R J, Ercole G, Martins E M. Computing the first eigenvalue of the p-Laplacian via the inverse power method. J Funct Anal, 2009, 257: 243–270

[2] Chen M F. Variational formulas and approximation theorems for the first eigenvalue in dimension one. Sci China Ser A, 2001, 44(4): 409–418

[3] Chen M F. Speed of stability for birth–death processes. Front Math China, 2010, 5(3): 379–515

[4] Chen M F. Efficient initials for computing the maximal eigenpair. Front Math China, 2016, 11(6): 1379–1418. A package based on the paper is available onhttps://cran.r-project.org/web/packages/EfficientMaxEigenpair/index.htmlA MatLab package see Chen’s homepage

[5] Chen M F. Efficient algorithm for principal eigenpair of discrete p-Laplacian. Front Math China, 2018, 13(3): 509–524

[6] Chen M F, Wang L D, Zhang Y H. Mixed eigenvalues of discrete p-Laplacian. Front Math China, 2014, 9(6): 1261–1292

[7] Ercole G. An inverse iteration method for obtaining q-eigenpairs of the p-Laplacian in a general bounded domain. arXiv: 1510.03941

[8] Hein M, Bühler T. An inverse power method for nonlinear eigenproblems with applications in 1-spectral clustering and sparse PCA. Advances in Neural Information Processing Systems, 2010, 23: 847–855

[9] Hynd R, Lindgren E. Inverse iteration for p-ground states. Proc Amer Math Soc, 2016, 144(5): 2121–2131

[10] Li Y S. The inverse iteration method for discrete weighted p-Laplacian. MS Thesis, Beijing Normal University, 2017 (in Chinese)

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2018 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature