Advertisement

Generation of point sets by convex optimization for interpolation in reproducing kernel Hilbert spaces

  • Ken’ichiro TanakaEmail author
Original Paper
  • 55 Downloads

Abstract

We propose algorithms to take point sets for kernel-based interpolation of functions in reproducing kernel Hilbert spaces (RKHSs) by convex optimization. We consider the case of kernels with the Mercer expansion and propose an algorithm by deriving a second-order cone programming (SOCP) problem that yields n points at one sitting for a given integer n. In addition, by modifying the SOCP problem slightly, we propose another sequential algorithm that adds an arbitrary number of new points in each step. Numerical experiments show that in several cases the proposed algorithms compete with the P-greedy algorithm, which is known to provide nearly optimal points.

Keywords

Reproducing kernel Hilbert space Kernel interpolation Point set Power function Optimal design Second-order cone programming 

Mathematics Subject Classification (2010)

65D05 65D15 41A05 46E22 

Notes

Acknowledgments

The author thanks Takanori Maehara for his valuable comment about the MATLAB programs used in the numerical experiments. Thanks to the comment, much faster execution of the proposed algorithms has been realized than that in the initially submitted version of this article. Furthermore, the author gives thanks to the anonymous reviewers for their valuable suggestions about this article.

Funding information

The author is supported by the grant-in-aid of Japan Society of the Promotion of Science with KAKENHI Grant Number 17K14241.

References

  1. 1.
    Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program., Ser. B 95, 3–51 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Atkinson, K., Han, W.: Spherical Harmonics and Approximations on the Unit Sphere: an Introduction. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  3. 3.
    Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization: Theory and Examples, 2nd edn. Springer, New York (2006)CrossRefzbMATHGoogle Scholar
  4. 4.
    Briani, M., Sommariva, A., Vianello, M.: Computing Fekete and Lebesgue points: simplex, square, disk. J. Comput. Appl. Math. 236, 2477–2486 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Buhmann, M.: Radial basis functions. Acta Numer. 10, 1–38 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dai, F., Xu, Y.: Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  7. 7.
    Fasshauer, G.: Meshfree Approximation Methods with MATLAB. World Scientific, Singapore (2007)CrossRefzbMATHGoogle Scholar
  8. 8.
    Fasshauer, G., McCourt, M.: Kernel-Based Approximation Methods Using MATLAB. World Scientific, Singapore (2015)CrossRefzbMATHGoogle Scholar
  9. 9.
    Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284, 193–228 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    De Marchi, S.: On optimal center locations for radial basis function interpolation: computational aspects. Rend. Sem. Mat. Univ. Pol. Torino 61, 343–358 (2003)MathSciNetzbMATHGoogle Scholar
  11. 11.
    De Marchi, S.: Geometric greedy and greedy points for RBF interpolation. In: Proceedings of the International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2009 (2009)Google Scholar
  12. 12.
    De Marchi, S., Schaback, R., Wendland, H.: Near-optimal data-independent point locations for radial basis function interpolation. Adv. Comput. Math. 23, 317–330 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nguyen, V.P., Rabczuk, T., Bordas, S., Duflot, M.: Meshless methods: a review and computer implementation aspects. Math. Comput. Simul. 79, 763–813 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Müller, S.: Complexity and stability of kernel-based reconstructions (in German). dissertation, Georg-August-Universität Göttingen, Institut für Numerische und Angewandte Mathematik, Lotzestrasse 16-18, D-37083 Göttingen, Jan 2009. Göttinger Online Klassifikation: EDDF 050 (2009)Google Scholar
  15. 15.
    Sangol, G.: Computing optimal designs of multiresponse experiments reduces to second order cone programming. J. Statist. Plann. Inference 141, 1684–1708 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Sangol, G., Harman, R.: Computing exact D-optimal designs by mixed integer second-order cone programming. Ann. Statist. 43, 2198–2224 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Santin, G., Haasdonk, B.: Convergence rate of the data-independent P-greedy algorithm in kernel-based approximation. Dolomites Research Notes on Approximation 10, 68–78 (2017)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Schaback, R., Wendland, H.: Adaptive greedy techniques for approximate solution of large RBF systems. Numer. Algor. 24, 239–254 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Schaback, R., Wendland, H.: Kernel techniques: from machine learning to meshless methods. Acta Numer. 15, 543–639 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Tanaka, K.: Matlab programs for generating point sets for interpolation in reproducing kernel Hilbert spaces. https://github.com/KeTanakaN/mat_points_interp_rkhs (last accessed on October 19, 2018)
  21. 21.
    Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyUniversity of TokyoBunkyo-kuJapan

Personalised recommendations