Advertisement

Constructive Approximation

, Volume 39, Issue 2, pp 323–341 | Cite as

Improved Exponential Convergence Rates by Oversampling Near the Boundary

  • Christian Rieger
  • Barbara Zwicknagl
Article

Abstract

Sampling inequalities for smooth functions bound a continuous norm in terms of a discretized norm and an error term that tends to zero exponentially as the discrete data set becomes dense. Improved estimates are derived for discrete point sets that cluster near the boundary, in particular for scattered point sets that are distributed quadratically in a boundary layer, and for tensorized Chebyshev grids. If applied to residuals of stable reconstruction processes, such inequalities yield exponential convergence orders. Our results agree with the observation that exponential deterministic approximation rates are often improved globally if the data sets are distributed more densely near the boundary.

Keywords

Sampling inequalities Convergence orders Boundary effect Smooth kernels Gaussian Inverse multiquadrics Chebyshev nodes 

Mathematics Subject Classification

41A05 41A25 41A63 65D05 

Notes

Acknowledgements

We thank R. Schaback, M. Griebel, and T. Hangelbroek for helpful discussions, and the referees for their careful reading and helpful comments. Some preliminary results are contained in C.R.’s PhD thesis defended at the University of Göttingen (see [27]). C.R. acknowledges partial support by the DFG through the Collaborative Research Centers (SFB) 611 and 1060, and the Hausdorff Center for Mathematics. B.Z. warmly thanks the Center for Nonlinear Analysis (NSF Grant No. DMS-0635983), where part of this research was carried out. Her research was partly funded by a postdoctoral fellowship of the National Science Foundation under Grant No. DMS-0905778.

References

  1. 1.
    Agadzhanov, A.N.: Functional properties of Sobolev spaces of infinite order. Sov. Math. Dokl. 38(1), 88–92 (1989) zbMATHMathSciNetGoogle Scholar
  2. 2.
    Arcangéli, R., López di 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(2), 181–211 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Arcangéli, R., López di Silanes, M.C., Torrens, J.J.: Estimates for functions in Sobolev spaces defined on unbounded domains. J. Approx. Theory 161(1), 198–212 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Arcangéli, R., López di Silanes, M.C., Torrens, J.J.: Extension of sampling inequalities to Sobolev semi-norms of fractional order and derivative data. Numer. Math. 121, 587–608 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Borwein, P., Erdelyi, T.: Polynomials and Polynomial Inequalities. Springer, New York (1995) CrossRefzbMATHGoogle Scholar
  6. 6.
    Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994) CrossRefzbMATHGoogle Scholar
  7. 7.
    Buhmann, M.D.: Multivariate cardinal interpolation with radial-basis functions. Constr. Approx. 6, 225–255 (1990) zbMATHMathSciNetGoogle Scholar
  8. 8.
    Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003) CrossRefGoogle Scholar
  9. 9.
    Buhmann, M.D., Powell, M.J.D.: Radial basis function interpolation on an infiniter egular grid. In: Mason, J.C., Cox, M.G. (eds.) Algorithms for Approximation II, pp. 146–169. Chapman and Hall, London (1990) CrossRefGoogle Scholar
  10. 10.
    Cheney, E.W.: An Introduction to Approximation Theory. McGraw-Hill, New York (1966) Google Scholar
  11. 11.
    De Vore, R.A., Lorentz, G.G.: Constructive Approximation. Grundlehren der mathematischen Wisenschaften. Springer, Berlin (1993) Google Scholar
  12. 12.
    Duchon, J.: Sur l’erreur d’ interpolation des fonctions de plusieurs variables par les Dm-splines. Rev. Française Automat. Informat. Rech. Opèr. Anal. Numer. 12, 325–334 (1978) zbMATHMathSciNetGoogle Scholar
  13. 13.
    Fornberg, B., Driscoll, T.A., Wright, G., Charles, R.: Observations on the behaviour of radial basis function approximation near boundaries. Comput. Math. Appl. 43, 473–490 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Hangelbroek, T.: Error estimates for thin plate spline approximation in the disk. Constr. Approx. 28, 27–59 (2008) zbMATHMathSciNetGoogle Scholar
  15. 15.
    Jetter, K., Stöckler, J., Ward, J.D.: Norming sets and scattered data approximation on spheres. In: Approximation Theory IX, Vol. II: Computational Aspects, pp. 137–144 (1998). Vanderbilt University Press Google Scholar
  16. 16.
    Johnson, M.J.: A bound on the approximation order of surface splines. Constr. Approx. 14, 429–438 (1998) zbMATHMathSciNetGoogle Scholar
  17. 17.
    Johnson, M.J.: The l p-approximation order of surface spline interpolation for 1≤p≤2. Constr. Approx. 20, 303–324 (2004) zbMATHMathSciNetGoogle Scholar
  18. 18.
    Krebs, J.: Support vector regression for the solution of linear integral equations. Inverse Probl. 27(6), 065007 (2011) CrossRefMathSciNetGoogle Scholar
  19. 19.
    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) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Madych, W.R.: An estimate for multivariate interpolation II. J. Approx. Theory 142, 116–128 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Madych, W.R., Nelson, S.A.: Multivariate interpolation and conditionally positive definite functions. J. Approx. Theory Appl. 4, 77–89 (1988) zbMATHMathSciNetGoogle Scholar
  22. 22.
    Madych, W.R., Nelson, S.A.: Multivariate interpolation and conditionally positive definite functions II. Math. Comput. 54, 211–230 (1990) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Madych, W.R., Nelson, S.A.: Bounds on multivariate polynomials and exponential error estimates for multi quadric interpolation. J. Approx. Theory 70, 94–114 (1992) CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    De Marchi, S., Schaback, R.: Stability of kernel-based interpolation. Adv. Comput. Math. 32, 155–161 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Matveev, O.: On a method for interpolating functions on chaotic nets. Math. Notes 62, 339–349 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Narcowich, F.J., Ward, J.D., Wendland, H.: Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting. Math. Comput. 74, 743–763 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Rieger, C.: Sampling inequalities and applications. PhD thesis, Universität Göttingen (2008), available online at http://hdl.handle.net/11858/00-1735-0000-0006-B3B9-0
  28. 28.
    Rieger, C., Schaback, R., Zwicknagl, B.: Sampling and stability. In: Mathematical Methods for Curves and Surfaces. Lecture Notes in Computer Science, vol. 5862, pp. 347–369. Springer, New York (2010) CrossRefGoogle Scholar
  29. 29.
    Rieger, C., Zwicknagl, B.: Deterministic error analysis of support vector machines and related regularized kernel methods. J. Mach. Learn. Res. 10, 2115–2132 (2009) zbMATHMathSciNetGoogle Scholar
  30. 30.
    Rieger, C., Zwicknagl, B.: Sampling inequalities for infinitely smooth functions, with applications to interpolation and machine learning. Adv. Comput. Math. 32(1), 103–129 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Schaback, R.: Comparison of radial basis function interpolants. In: Jetter, K., Schumaker, L.L., Utreras, F. (eds.) Multivariate Approximations: From CAGD to Wavelets, pp. 293–305. World Scientific, Singapore (1993) CrossRefGoogle Scholar
  32. 32.
    Schaback, R., Wendland, H.: Kernel techniques: from machine learning to meshless methods. Acta Numer. 15, 543–639 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Wendland, H.: Local polynomial reproduction and moving least squares approximation. IMA J. Numer. Anal. 21, 285–300 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Wendland, H.: Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2005) zbMATHGoogle Scholar
  35. 35.
    Wendland, H., Rieger, C.: Approximate interpolation with applications to selecting smoothing parameters. Numer. Math. 101, 729–748 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Zhang, X., Jiang, X.: Numerical analyses of the boundary effect of radial basis functions in 3d surface reconstruction. Numer. Algorithms 47, 327–339 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Zwicknagl, B.: Power series kernels. Constr. Approx. 29, 61–84 (2009) zbMATHMathSciNetGoogle Scholar
  38. 38.
    Zwicknagl, B., Schaback, R.: Interpolation and approximation in Taylor spaces. J. Approx. Theory 171, 65–83 (2013) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute for Numerical SimulationUniversity of BonnBonnGermany
  2. 2.Institute for Applied MathematicsUniversity of BonnBonnGermany

Personalised recommendations