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Efficient algorithm for principal eigenpair of discrete p-Laplacian |
Mu-Fa CHEN() |
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems (Beijing Normal University), Ministry of Education, Beijing 100875, China |
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Abstract This paper is a continuation of the author’s previous papers [Front. Math. China, 2016, 11(6): 1379–1418; 2017, 12(5): 1023–1043], where the linear case was studied. A shifted inverse iteration algorithm is introduced, as an acceleration of the inverse iteration which is often used in the non-linear context (the p-Laplacian operators for instance). Even though the algorithm is formally similar to the Rayleigh quotient iteration which is well-known in the linear situation, but they are essentially different. The point is that the standard Rayleigh quotient cannot be used as a shift in the non-linear setup. We have to employ a different quantity which has been obtained only recently. As a surprised gift, the explicit formulas for the algorithm restricted to the linear case (p = 2) is obtained, which improves the author’s approximating procedure for the leading eigenvalues in different context, appeared in a group of publications. The paper begins with p-Laplacian, and is closed by the non-linear operators corresponding to the well-known Hardy-type inequalities.
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Keywords
Discrete p-Laplacian
principal eigenpair
shifted inverse iteration
approximating procedure
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Corresponding Author(s):
Mu-Fa CHEN
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Issue Date: 11 June 2018
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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
https://doi.org/10.1016/j.jfa.2009.01.023
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2 |
Chen M F. Speed of stability for birth–death processes. Front Math China, 2010, 5(3): 379–515
https://doi.org/10.1007/s11464-010-0068-7
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3 |
Chen M F. The optimal constant in Hardy-type inequalities. Acta Math Sin (Engl Ser), 2015, 31(5): 731–754
https://doi.org/10.1007/s10114-015-4731-5
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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 on CRAN now. One may check it through the link: A MatLab package is also available, see the author’s homepage
https://doi.org/10.1007/s11464-016-0573-4
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A MatLab package is also available, see the author’s homepage
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Chen M F. Global algorithms for maximal eigenpair. Front Math China, 2017, 12(5): 1023–1043
https://doi.org/10.1007/s11464-017-0658-8
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7 |
Chen M F, Wang L D, Zhang Y H. Mixed eigenvalues of discrete p-Laplacian. Front Math China, 2014, 9(6): 1261–1292
https://doi.org/10.1007/s11464-014-0374-6
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8 |
Ercole G. An inverse iteration method for obtaining q-eigenpairs of the p-Laplacian in a general bounded domain. Mathematics, 2015
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9 |
Li Y S. The inverse iteration method for discrete weighted p-Laplacian. Master’s Thesis, Beijing Normal University, 2017 (in Chinese)
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10 |
Liao Z W. Discrete weighted Hardy inequalities with different kinds of boundary conditions. Acta Math Sin (Engl Ser), 2016, 32(9): 993–1013
https://doi.org/10.1007/s10114-016-5675-0
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