Advertisement

Numerische Mathematik

, Volume 137, Issue 3, pp 579–605 | Cite as

Radial basis function approximation of noisy scattered data on the sphere

  • Kerstin HesseEmail author
  • Ian H. Sloan
  • Robert S. Womersley
Article
  • 409 Downloads

Abstract

In this paper we consider the approximation of noisy scattered data on the sphere by radial basis functions generated by a strictly positive definite kernel. The approximation is the minimizer in the native space for that kernel of a quadratic functional in which the smoothing term is a multiple of the square of the native space norm. The balance between data fitting and smoothness is controlled by a smoothing parameter, the choice of which should depend on the nature and magnitude of the noise. The main results concern the choice of that smoothing parameter, under the assumption that the noise is deterministic rather than random. Four strategies for choosing the smoothing parameter are considered: Morozov’s discrepancy principle, and three a priori strategies. For each of these strategies we derive an \(L_2\) error bound. The error bounds are similar, with the discrepancy principle giving marginally the best bound. A numerical example supports the theoretical results.

Mathematics Subject Classification

Primary 41A15 65D07 65D10 Secondary 33C55 42C10 

Notes

Acknowledgements

The authors are particularly grateful to one of the anonymous referees who pointed out that Theorem 4.1 and its proof could be simplified and improved. We also owe this referee the a priori parameter choice (iii) in Theorem 4.1.

References

  1. 1.
    Alfeld, P., Neamtu, M., Schumaker, L.L.: Fitting scattered data on sphere-like surfaces using spherical splines. J. Comput. Appl. Math. 73, 5–43 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    An, C., Chen, X., Sloan, I.H., Womersley, R.S.: Regularized least squares approximations on the sphere using spherical designs. SIAM J. Numer. Anal. 50(3), 1513–1534 (2012). (Corrigendum: SIAM J. Numer. Anal. 52(4), 2205–2206 (2014).)Google Scholar
  3. 3.
    Arcangéli, R., López de Silanes, M.C., Torrens, J.J.: An extension of a bound for functions in Sobolev spaces, with applications to \((m, s)\)-spline interpolation and smoothing. Numer. Math. 107, 181–211 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bauer, F., Lukas, M.A.: Comparing parameter choice methods for regularization of ill-posed problems. Math. Comput. Simul. 81, 1795–1841 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, D., Menegatto, V.A., Sun, X.: A necessary and sufficient condition for strictly positive definite functions on spheres. Proc. Am. Math. Soc. 131, 2733–2740 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cucker, F., Zhou, D.X.: Learning Theory: An Approximation Theory Viewpoint. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar
  7. 7.
    Erdélyi, A. (ed.), Magnus, W., Oberhettinger, F., Tricomi, F.G. (research associates): Higher Transcendental Functions, Vol. III. California Institute of Technology, Bateman Manuscript Project, McGraw-Hill Book Company Inc, New York (1953)Google Scholar
  8. 8.
    Freeden, W.: Multiscale Modelling of Spaceborne Geodata. B. G. Teubner, Stuttgart (1999)zbMATHGoogle Scholar
  9. 9.
    Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere (with Applications to Geomathematics). Clarendon Press, Oxford (1998)zbMATHGoogle Scholar
  10. 10.
    Hangelbroek, T., Narcowich, F.J., Sun, X., Ward, J.D.: Kernel approximation on manifolds II: the \(L_\infty \) norm of the \(L_2\) projector. SIAM J. Math. Anal. 43(2), 662–684 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hangelbroek, T., Narcowich, F.J., Ward, J.D.: Kernel approximation on manifolds I: bounding the Lebesgue constant. SIAM J. Math. Anal. 42(4), 1732–1760 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hansen, P.C.: Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34(4), 561–580 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. Springer, New York (1996)CrossRefzbMATHGoogle Scholar
  14. 14.
    Krebs, J., Louis, A.K., Wendland, H.: Sobolev error estimates and a priori parameter selection for semi-discrete Tikhonov regularization. J. Inverse Ill Posed Probl. 17, 845–869 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Le Gia, Q.T., Narcowich, F.J., Ward, J.D., Wendland, H.: Continuous and discrete least-squares approximation by radial basis functions on spheres. J. Approx. Theory 143, 124–133 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1. Springer, Berlin (1972)CrossRefzbMATHGoogle Scholar
  17. 17.
    McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  18. 18.
    Mhaskar, H.M., Narcowich, F.J., Ward, J.D.: Spherical Marcinkiewicz–Zygmund inequalities. Math. Comput. 70(235), 1113–1130 (2001). (Corrigendum: Math. Comput. 71, 453–454 (2002).)Google Scholar
  19. 19.
    Narcowich, F.J., Ward, J.D.: Scattered data interpolation on spheres: error estimates and locally supported basis functions. SIAM J. Math. Anal. 33(6), 1393–1410 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Reimer, M.: Multivariate Polynomial Approximation. Birkhäuser, Basel (2003)CrossRefzbMATHGoogle Scholar
  21. 21.
    Schoenberg, I.J.: Positive definite functions on spheres. Duke Math. J. 9, 96–108 (1942)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sloan, I.H., Womersley, R.S.: Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math. 21, 107–125 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Szegö, G.: Orthogonal Polynomials, vol. XXIII, 4th edn. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence (1975)zbMATHGoogle Scholar
  24. 24.
    Vapnik, V.N.: Statistical Learning Theory. Wiley, New York (1998)zbMATHGoogle Scholar
  25. 25.
    von Golitschek, M., Schumaker, L.L.: Data fitting by penalized least squares. In: Algorithms for Approximation. II (Shrivenham, 1988), pp. 210–227. Chapman and Hall, London (1990)Google Scholar
  26. 26.
    Wahba, G.: Spline models for observational data. In: CBMS-NSF regional conference series in applied mathematics, vol. 59, SIAM, Philadelphia (1990)Google Scholar
  27. 27.
    Wei, T., Hon, Y.C., Wang, Y.B.: Reconstruction of numerical derivatives from scattered noisy data. Inverse Problems 21, 657–672 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4, 389–396 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wendland, H.: Error estimates for interpolation by compactly supported radial basis functions of minimal degree. J. Approx. Theory 93, 258–272 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar
  31. 31.
    Xu, Y., Cheney, E.W.: Strictly positive definite functions on spheres. Proc. Am. Math. Soc. 116, 977–981 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Zhang, Y., Li, R., Tsai, C.-L.: Regularization parameter selections via generalized information criterion. J. Am. Stat. Assoc. 105(489), 312–323 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsPaderborn UniversityPaderbornGermany
  2. 2.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

Personalised recommendations