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Advances in Computational Mathematics

, Volume 41, Issue 1, pp 23–53 | Cite as

Order-preserving derivative approximation with periodic radial basis functions

  • Edward Fuselier
  • Grady B. Wright
Article

Abstract

In this exploratory paper we study the convergence rates of an iterated method for approximating derivatives of periodic functions using radial basis function (RBF) interpolation. Given a target function sampled on some node set, an approximation of the m th derivative is obtained by m successive applications of the operator “interpolate, then differentiate” - this process is known in the spline community as successive splines or iterated splines. For uniformly spaced nodes on the circle, we give a sufficient condition on the RBF kernel to guarantee that, when the error is measured only at the nodes, this iterated method approximates all derivatives with the same rate of convergence. We show that thin-plate spline, power function, and Matérn kernels restricted to the circle all satisfy this condition, and numerical evidence is provided to show that this phenomena occurs for some other popular RBF kernels. Finally, we consider possible extensions to higher-dimensional periodic domains by numerically studying the convergence of an iterated method for approximating the surface Laplace (Laplace-Beltrami) operator using RBF interpolation on the unit sphere and a torus.

Keywords

Numerical differentiation Periodic radial basis functions Circular basis functions Iterated splines Superconvergence 

Mathematical Subject Classifications (2010)

41A05 41A25 41A30 65D25 

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References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. In: National Bureau of Standards Applied Mathematics Series, vol. 55. For Sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC (1964)Google Scholar
  2. 2.
    Baxter, B.J.C., Hubbert, S.: Radial basis functions for the sphere. In: Recent Progress in Multivariate Approximation (Written-B ommerholz, 2000), International Series Numerical Mathematics, vol. 137, pp. 33–47. Basel, Birkhäuser (2001)Google Scholar
  3. 3.
    Delvos, F.-J.: Approximation properties of periodic interpolation by translates of one function. RAIRO Modél. Math. Anal. Numér. 28(2), 177–188 (1994)MATHMathSciNetGoogle Scholar
  4. 4.
    Fasshauer, G.E.: Meshfree approximation methods with MATLAB. In: Interdisciplinary Mathematical Sciences, vol. 6. World Scientific Publishing Co. Pte. Ltd., Hackensack. With 1 CD-ROM (Windows, Macintosh and UNIX) (2007)Google Scholar
  5. 5.
    Fornberg, B., Piret, C.: A stable algorithm for flat radial basis functions on a sphere. SIAM J. Sci. Comp. 30, 60–80 (2007)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Fuselier, E.J.: Nodes used in order-preserving approximation of derivatives with periodic radial basis functions. Accessed 2012. http://math.highpoint.edu/~efuselier/OrderPreservingData/
  7. 7.
    Fuselier, E.J., Wright, G.B.: Scattered data interpolation on embedded submanifolds with restricted positive definite kernels: sobolev error estimates. SIAM J. Numer. Anal. 50(3), 1753–1776 (2012)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Fuselier, E.J., Wright, G.B.: A high-order kernel method for diffusion and reaction-diffusion equations on surfaces. J. Sci. Comput. 56(3), 535–565 (2013)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Lê Gia, Q.T.: Approximation of parabolic pdes on spheres using spherical basis functions. Adv. Comput. Math. 22, 377–397 (2005)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Golomb, M.: Approximation by periodic spline interpolants on uniform meshes. J. Approx. Theory 1, 26–65 (1968)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    González, Á.: Measurement of areas on a sphere using Fibonacci and latitude-longitude lattices. Math. Geosci. 42(1), 49–64 (2010)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Hubbert, S., Müller, S.: Interpolation with circular basis functions. Numer. Algoritm. 42(1), 75–90 (2006)CrossRefMATHGoogle Scholar
  13. 13.
    Levesley, J., Kushpel, A.K.: Generalised sk-spline interpolation on compact abelian groups. J. Approx. Theory 97(2), 311–333 (1999)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Light, W.A., Cheney, E.W.: Interpolation by periodic radial basis functions. J. Math. Anal. Appl. 168(1), 111–130 (1992)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Lorentz, R.A., Narcowich, F.J., Ward, J.D.: Collocation discretizations of the transport equation with radial basis functions. Appl. Math. Comput. 145(1), 97–116 (2003)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Müller, C.: Spherical Harmonics, of Lecture Notes in Mathematics, vol. 17. Springer-Verlag, Berlin (1966)Google Scholar
  17. 17.
    Narcowich, F.J., Sun, X., Ward, J.D.: Approximation power of RBFs and their associated SBFs: a connection. Adv. Comput. Math. 27(1), 107–124 (2007)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Ridley, J.N.: Ideal phyllotaxis on general surfaces of revolution. Math. Biosci. 79(1), 1–24 (1986)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Shelley, M.J., Baker, G.R.: Order-preserving approximations to successive derivatives of periodic functions by iterated splines. SIAM J. Numer. Anal. 25(6), 1442–1452 (1988)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Stöckler. J.: Multivariate Bernoulli splines and the periodic interpolation problem. Constr. Approx. 7(1), 105–122 (1991)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Watson, G.N.: A treatise on the theory of Bessel functions. In: Cambridge Mathematical Library. Cambridge University Press, Cambridge (1995). Reprint of the second (1944) editionGoogle Scholar
  22. 22.
    Wendland, H.: Scattered Data Approximation, of Cambridge Monographs on Applied and Computational Mathematics, vol. 17. Cambridge University Press, Cambridge (2005)Google Scholar
  23. 23.
    Womersley, R.S., Sloan, I.H.: Interpolation and Cubature on the Sphere. Accessed 2012. http://web.maths.unsw.edu.au/rsw/Sphere/
  24. 24.
    Wright, G.B., Flyer, N., Yuen, D.: A hybrid radial basis function - pseudospectral method for thermal convection in a 3D spherical shell. Geochem. Geophys. Geosyst. 11, Q07003 (2010)CrossRefGoogle Scholar
  25. 25.
    Xu, Y., Cheney, E.W.: Strictly positive definite functions on spheres. Proc. Am. Math. Soc. 116(4), 977–981 (1992)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    zu Castell, W., Filbir, F.: Radial basis functions and corresponding zonal series expansions on the sphere. J. Approx. Theory 134(1), 65–79 (2005)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceHigh Point UniversityHigh PointUSA
  2. 2.Department of MathematicsBoise State UniversityBoiseUSA

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