Effective condition number for the selection of the RBF shape parameter with the fictitious point method

Based on the Uncertainty Principle of radial basis functions (RBFs), it is known that the condition number and the error cannot be both kept small at the same time. In contrast to the traditional condition number, the effective condition number provides a much better estimation of the actual conditi...

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Bibliographic Details
Published in:Applied numerical mathematics 2022-08, Vol.178, p.280-295
Main Authors: Noorizadegan, Amir, Chen, Chuin-Shan, Young, D.L., Chen, C.S.
Format: Article
Language:English
Subjects:
Collocation
Effective condition number
Fictitious point method
Kansa method
Multiquadrics
Radial basis functions
Shape parameter
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Summary:Based on the Uncertainty Principle of radial basis functions (RBFs), it is known that the condition number and the error cannot be both kept small at the same time. In contrast to the traditional condition number, the effective condition number provides a much better estimation of the actual condition number of the resultant matrix system. In this paper, motivated by the Uncertainty Principle of RBFs, we propose to apply the effective condition number as a numerical tool to determine a reasonably good shape parameter value in the context of the Kansa method coupled with the fictitious point method. Six examples for second and fourth order partial differential equations in 2D and 3D are presented to demonstrate the effectiveness of the proposed method.
ISSN:0168-9274
1873-5460
DOI:10.1016/j.apnum.2022.04.003