Advertisement

Advances in Computational Mathematics

, Volume 3, Issue 3, pp 251–264 | Cite as

Error estimates and condition numbers for radial basis function interpolation

  • Robert Schaback
Article

Abstract

For interpolation of scattered multivariate data by radial basis functions, an “uncertainty relation” between the attainable error and the condition of the interpolation matrices is proven. It states that the error and the condition number cannot both be kept small. Bounds on the Lebesgue constants are obtained as a byproduct. A variation of the Narcowich-Ward theory of upper bounds on the norm of the inverse of the interpolation matrix is presented in order to handle the whole set of radial basis functions that are currently in use.

Keywords

Radial Basis Function Lebesgue Constant Interpolation Matrix Radial Basis Function Interpolation Radial Basis Func 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [9]
    W.R. Madych and S.A. Nelson, Multivariate interpolation: a variational theory, Manuscript (1983).Google Scholar
  2. [10]
    W.R. Madych and S.A. Nelson, Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation. J. Approx. Theory 70 (1992) 94–114.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [11]
    F.J. Narcowich and J.D. Ward, Norm of inverses and condition numbers for matrices associated with scattered data. J. Approx. Theory 64 (1991) 69–94.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [12]
    F.J. Narcowich and J.D. Ward, Norms of inverses for matrices associated with scattered data, in:Curves and Surfaces, eds. P.J. Laurent, A. Le Méhauté and L.L. Schumaker (Academic Press, Boston, 1991) pp. 341–348.Google Scholar
  5. [13]
    F.J. Narcowich and J.D. Ward, Norm estimates for the inverses of a general class of scattered-data radial-function interpolation matrices, J. Approx. Theory 69 (1992) 84–109.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [14]
    M.J.D. Powell, Univariate multiquadric interpolation: Some recent results, in:Curves and Surfaces, eds. P.J. Laurent, A. Le Méhauté and L.L. Schumaker (Academic Press, 1991) pp. 371–382.Google Scholar
  7. [15]
    R. Schaback, Comparison of radial basis function interpolants, in:Multivariate Approximation: From CAGD to Wavelets, eds. K. Jetter and F. Utreras, (World Scientific, London, 1993) pp. 293–305.Google Scholar
  8. [16]
    R. Schaback, Lower bounds for norms of inverses of interpolation matrices for radial basis functions, J. Approx. Theory 79 (1994) 287–306.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [17]
    X. Sun, Norm estimates for inverses of Euclidean distance matrices, J. Approx. Theory 70 (1992) 339–347.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [18]
    Z. Wu and R. Schaback, Local error estimtes for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993) 13–27.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [14]
    H.P. Seidel, Symmetric recursive algorthms for curves, Comp. Aided Geom. Design 7 (1990) 57–67.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [15]
    H.P. Seidel, Polar forms for geometrically continuous spline curves of arbitrary degree, ACM Trans. Graphics 12 (1993) 1–34.CrossRefzbMATHGoogle Scholar
  13. [16]
    K. Strøm, Splines, polynomials and polar forms. Ph.D. dissertation, University of Oslo, Norway (1992).Google Scholar
  14. [17]
    K. Strøm, Products of B-patches, Numer. Algor. 4 (1993) 323–337.CrossRefGoogle Scholar

References

  1. [1]
    A.S. Cavaretta, W. Dahmen and C.A. Micchelli,Stationary Subdivision, Memoirs of Amer. Math. Soc., Vol. 93 (1991).Google Scholar
  2. [2]
    D. Colella and C. Heil, Characterizations of scaling functions: continuous solutions, SIAM J. Matrix Anal. Appl. 15 (1994) 496–518.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    W. Dahmen and C.A. Micchelli, Translates of multivariate splines, Lin. Alg. Appl. 52 (1983) 217–234.MathSciNetGoogle Scholar
  4. [4]
    G. Deslauriers and S. Dubuc, Symmetric iterative interpolation process, Constr. Approx. 5 (1989) 49–68.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    I. Daubechies and J.C. Lagarias, Two-scale difference equations: I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991) 1388–1410.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    I. Daubechies and J.C. Lagarias, Two-scale difference equations: II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal. 23 (1992) 1031–1079.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    R.A. DeVore and G.G. Lorentz,Constructive Approximation (Springer, Berlin, 1993).CrossRefzbMATHGoogle Scholar
  8. [8]
    S. Dubuc, Interpolation through an iterative scheme, J. Math. Anal. Appl. 114 (1986) 185–205.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    N. Dyn, Subdivision schemes in computer aided geometric design, in:Advances in Numerical Analysis II — Wavelets, Subdivision Algorithms and Radius Functions, ed. W.A. Light (Clarendon Press, Oxford, 1991) pp. 36–104.Google Scholar
  10. [10]
    N. Dyn, J.A. Gregory and D. Levin, Analysis of uniform binary subdivision schemes for curve design. Constr. Approx. 7 (1991) 127–147.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    T. Eirola, Sobolev characterization of solutions of dilation equations, SIAM J. Math. Anal. 23 (1992) 1015–1030.MathSciNetCrossRefzbMATHGoogle Scholar

References

  1. [1]
    J.R. Dormand and P.J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math. 6 (1980) 19–26.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    K. Gustafsson, Control theoretic techniques for stepsize selection in explicit Runge-Kutta methods, ACM Trans. Math. Software 17 (1991) 533–544.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    K. Gustafsson, M. Lundh and G. Söderlind, A PI stepsize control for the numerical solution of ordinary differential equations, BIT 28 (1988) 270–287.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    E. Hairer and G. Wanner,Solving Ordinary Differential Equations II (Springer, Berlin, 1991).CrossRefzbMATHGoogle Scholar
  5. [5]
    G. Hall, Equilibrium states of Runge-Kutta formulae, ACM Trans. Math. Software 11 (1985) 289–301.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    G. Hall and D.J. Higham, Analysis of stepsize selection schemes for Runge-Kutta codes, IMA J. Numer. Anal. 8 (1988) 305–310.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    D.J. Higham and G. Hall, Embedded Runge-Kutta formulae with stable equilibrium states, J. Comput. Appl. Math. 29 (1990) 25–33.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    F.T. Krogh, On testing a subroutine for the numerical integration of ordinary differential equations, J. ACM 4 (1973) 545–562.CrossRefGoogle Scholar
  9. [9]
    P.J. Prince and J.R. Dormand, High order embedded Runge-Kutta formulae, J. Comput. Appl. Math. 7 (1981) 67–75.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    B.C. Robertson, Detecting stiffness with explicit runge-Kutta formulas, Report 193/87, Dept. Comp. Sci., University of Toronto (1987).Google Scholar
  11. [11]
    L.F. Shampine, Lipschitz constants and robust ODE codes, Technical Report SAND79-0458, Sandia National Laboratories, Albuquerque, New Mexico (March 1979).Google Scholar

References

  1. [1]
    M. Abramowitz and I.A. Stegun,Pocketbook of Mathematical Functions (Harri Deutsch, Thun, 1984).zbMATHGoogle Scholar
  2. [2]
    K. Ball, N. Sivakumar and J.D. Ward, On the sensitivity of radial basis interpolation to minimal data separation distance, Constr. Approx. 8 (1992) 401–426.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    C. de Boor, The quasi-interpolant as a tool in elementary polynomial spline theory in:Approximation Theory, ed. G.G. Lorentz (Academic Press, New York, 1973) pp. 269–276.Google Scholar
  4. [4]
    C. de Boor,A Practical Guide to Splines (Springer, New York, 1978).CrossRefzbMATHGoogle Scholar
  5. [5]
    C. de Boor and G.J. Fix, Spline approximation by quasiinterpolants, J. Approx. Theory 8 (1973) 19–45.CrossRefzbMATHGoogle Scholar
  6. [6]
    M.D. Buhmann, Discrete least squares approximation and prewavelets from radial function spaces, Math. Proc. Cambridge Phil. Soc. 114 (1993) 533–558.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M.D. Buhmann and C.A. Micchelli, Spline prewavelets for non-uniform knots, Numer. Math. 61 (1992) 455–474.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    C.K. Chui, K. Jetter, J. Stöckler and J.D. Ward, Wavelets for analyzing scattered data: An unbounded operator approach, ms. (November 1994).Google Scholar
  9. [9]
    C.K. Chui, K. Jetter and J.D. Ward, Cardinal interpolation with differences of tempered functions, Comp. Math. Appl. 24 (1992) 35–48.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    I. Daubechies,Ten Lectures on Wavelets, CBMS-NSF Reg. Conf. Series in Appl. Math., vol. 61 (SIAM, Philadelphia, 1992).CrossRefzbMATHGoogle Scholar
  11. [11]
    R.A. DeVore and G.G. Lorentz,Constructive Approximation (Springer, New York, 1994).Google Scholar
  12. [12]
    I.M. Gel’fand and G.E. Shilov,Generalized Functions, vol. 1 (Academic Press, New York, 1964).zbMATHGoogle Scholar
  13. [13]
    I.M. Gelfand and N.Ya. Vilenkin,Generalized Functions, vol. 4 (Academic Press, New York, 1964).Google Scholar
  14. [14]
    M.J.D. Powell, Univariate multiquadric approximation: Reproduction of linear polynomials, in:Multivariate Approximation and Interpolation, eds. W. Hau\mann and K. Jetter, ISNM 94 (Birkhäuser, Basel, 1990) pp. 227–240.CrossRefGoogle Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1995

Authors and Affiliations

  • Robert Schaback
    • 1
  1. 1.Institut für Numerische und Angewandte MathematikUniversität GöttingenGöttingenGermany

Personalised recommendations