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Advances in Computational Mathematics

, Volume 11, Issue 2–3, pp 183–192 | Cite as

Proof of convergence of an iterative technique for thin plate spline interpolation in two dimensions

  • A.C. Faul
  • M.J.D. Powell
Article

Abstract

Thin plate spline methods provide an interpolant to values of a real function and are highly useful in many applications. The need for iterative procedures arises, since hardly any sparsity occurs in the linear system of interpolation equations. This paper considers a generalization of the iterative algorithm developed by Beatson, Goodsell and Powell. A proof of convergence of this method is given. It depends mainly on the remark that all the changes to the thin plate spline coefficients reduce a certain semi‐norm of the difference between the required interpolant and the current approximation. The analysis applies also to analogous algorithms for other radial basis functions.

Keywords

Radial Basis Function Iterative Algorithm Interpolation Point Iterative Technique Thin Plate Spline 
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.

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References

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    R.K. Beatson, G. Goodsell and M.J.D. Powell, On multigrid techniques for thin plate spline interpolation in two dimensions, in: The Mathematics of Numerical Analysis, eds. J. Renegar, M. Shub and S. Smale (Amer. Math. Soc., Providence, RI, 1995) pp. 77-97.Google Scholar
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    C.A. Micchelli, Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. Approx. 2 (1986) 11-22.zbMATHMathSciNetCrossRefGoogle Scholar
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    M.J.D. Powell, A new iterative algorithm for thin plate spline interpolation in two dimensions, Ann. Numer. Math. 4 (1997) 519-527.zbMATHMathSciNetGoogle Scholar
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    M.J.D. Powell, The theory of radial basis function approximation in 1990, in: Advances in Numerical Analysis, Vol. II: Wavelets, Subdivision Algorithms, and Radial Basis Functions, ed. W.A. Light (Clarendon Press, Oxford, 1992) pp. 105-210.Google Scholar
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    R. Schaback, Comparison of radial basis function interpolants, in: Multivariate Approximations: From CAGD to Wavelets, eds. K. Jetter and F. Utreras (World Scientific, Singapore, 1993) pp. 293-305.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • A.C. Faul
    • 1
  • M.J.D. Powell
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeEnglandUK

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