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A Multiscale RBF Method for Severely Ill-Posed Problems on Spheres

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Abstract

We propose and analyze the support vector approach to approximating the solution of a severely ill-posed problem \(Au=f\) on the sphere, in which A is an ill-posed map from the unit sphere to a concentric larger sphere. The Vapnik’s \(\varepsilon \)-intensive function is adopted in the regularization technique to reduce the error induced by noisy data. The method is then extended to a multiscale algorithm by varying the support radius of the radial basis functions at each scale. We discuss the convergence of the multiscale support vector approach and provide strategies for choosing both regularization parameters and cut-off parameters at each level. Numerical examples are constructed to verify the efficiency of the multiscale support vector approach.

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Acknowledgements

M. Zhong is supported by the NSFC (No. 11871149) and supported by Zhishan Youth Scholar Program of SEU. The support from the Australian Research Council Discovery Grant DP180100506 is gratefully acknowledged.

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Author notes

  1. Quoc Thong Le Gia and Ian Hugh Sloan have contributed equally to this work.

Authors and Affiliations

  1. School of Mathematics, Southeast University, Nanjing, 210096, Jiangsu, People’s Republic of China

    Min Zhong

  2. Nanjing Center for Applied Mathematics, Nanjing, 211135, Jiangsu, People’s Republic of China

    Min Zhong

  3. School of Mathematics and Statistics, University of New South Wales, Kensington, Sydney, 2052, NSW, Australia

    Quoc Thong Le Gia & Ian Hugh Sloan

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  1. Min Zhong
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Correspondence to Min Zhong.

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Appendix

Appendix

We also illustrate that the quadratic program (41) is solved by MATLAB codes quadprog, the corresponding matrices are defined via (16), but modified into multiscale version.

figure b

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Zhong, M., Gia, Q.T.L. & Sloan, I.H. A Multiscale RBF Method for Severely Ill-Posed Problems on Spheres. J Sci Comput 94, 22 (2023). https://doi.org/10.1007/s10915-022-02046-9

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  • DOI: https://doi.org/10.1007/s10915-022-02046-9

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