Abstract
The goal of this paper is to construct data-independent optimal point sets for interpolation by radial basis functions. The interpolation points are chosen to be uniformly good for all functions from the associated native Hilbert space. To this end we collect various results on the power function, which we use to show that good interpolation points are always uniformly distributed in a certain sense. We also prove convergence of two different greedy algorithms for the construction of near-optimal sets which lead to stable interpolation. Finally, we provide several examples.
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References
R.K. Beatson, W.A. Light and S. Billings, Fast solution of the radial basis function interpolation equations: Domain decomposition methods, SIAM J. Sci. Comput. 22 (2000) 1717–1740.
A. Beyer, Optimale Centerverteilung bei Interpolation mit radialen Basisfunktionen, Diplomarbeit, Universität Göttingen (1994).
L.P. Bos and U. Maier, On the asymptotics of points which maximize determinants of the form det(g(|xi−xj|)), in: Advances in Multivariate Approximation, eds. W. Haussmann, K. Jetter and M. Reimer, Mathematical Research, Vol. 107 (Wiley/VCH, Berlin, 1999) pp. 1–22.
A. Iske, Optimal distribution of centers for radial basis function methods, Technical Report M0004, Technische Universität München (2000).
R. Schaback, Error estimates and condition numbers for radial basis function interpolation, Adv. Comput. Math. 3 (1995) 251–264.
R. Schaback, Multivariate interpolation and approximation by translates of a basis function, in: Approximation Theory VIII, eds. C. Chui and L. Schumaker, Approximation and Interpolation, Vol. 1 (World Scientific, Singapore, 1995) pp. 491–514.
R. Schaback, Native Hilbert spaces for radial basis functions I, in: New Developments in Approximation Theory, eds. M. Buhmann, D.H. Mache, M. Felten and M. Müller, International Series of Numerical Mathematics, Vol. 132 (Birkhäuser, Basel, 1999) pp. 255–282.
H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory 93 (1998) 258–272.
Z. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993) 13–27.
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Communicated by J. Ward
AMS subject classification
41A05, 41063, 41065, 65D05, 65D15
This work has been done with the support of the Vigoni CRUI-DAAD programme, for the years 2001/2002, between the Universities of Verona and Göttingen.
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De Marchi, S., Schaback, R. & Wendland, H. Near-optimal data-independent point locations for radial basis function interpolation. Adv Comput Math 23, 317–330 (2005). https://doi.org/10.1007/s10444-004-1829-1
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DOI: https://doi.org/10.1007/s10444-004-1829-1