Abstract
We propose a modified bivariate Shepard operator of least squares thin-plate spline type based on an iterative method, introduced by A.V. Masjukov and V.V. Masjukov. This iterative method does not require an artificial setup of the parameters and it is based on successive scaling. For the numerical examples, using an idea of J. R. McMahon, we construct some representative sets of knot points for the initial sets of the interpolation nodes.
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Paper coordinates
A. Malina, Iterative Shepard operator of least squares thin-plate spline type, Dolomites Res. Notes Approx., 16(2023), No. 3, pp. 57-62.
About this paper
Journal
Dolomites Research Notes on Approximation
Publisher Name
Padova University Press
DOI
10.14658/PUPJ-DRNA-2023-3-8
Online ISSN
2035-6803
Google Scholar Profile
[1] T. Cătinaș. The combined Shepard-Abel-Goncharov univariate operator. Rev. Anal. Numér. Théor. Approx., 32:11–20, 2003.
[2] T. Cătinaș. The combined Shepard-Lidstone bivariate operator. In: de Bruin, M.G. et al. (eds.): Trends and Applications in Constructive
Approximation. International Series of Numerical Mathematics, Springer Group-Birkhäuser Verlag, 151:77–89, 2005.
[3] T. Cătinaș. Bivariate interpolation by combined Shepard operators. Proceedings of 17th IMACS World Congress, Scientific Computation,
Applied Mathematics and Simulation, 7pp., 2005.
[4] T. Cătinaș. The bivariate Shepard operator of Bernoulli type. Calcolo, 44(4):189–202, 2007.
[5] T. Cătinaș. An iterative modification of Shepard-Bernoulli Operator. Results Math., 69:387–395, 2016.
[6] T. Cătinaș., A. Malina. Shepard operator of least squares thin-plate spline type. Stud. Univ. Babe¸s-Bolyai Math. 66(2):257–265, 2021.
[7] Gh. Coman. Hermite-type Shepard operators. Rev. Anal. Numér. Théor. Approx., 26:33–38, 1997.
[8] Gh. Coman. Shepard operators of Birkhoff type. Calcolo, 35:197–203, 1998.
[9] R. Franke. Scattered data interpolation: tests of some methods. Math. Comp., 38:181–200, 1982.
[10] R. Franke, G. Nielson. Smooth interpolation of large sets of scattered data. Int. J. Numer. Meths. Engrg. 15:1691–1704, 1980.
[11] A.V. Masjukov, V.V. Masjukov. Multiscale modification of Shepard’s method for interpolation of multivariate scattered data. Mathematical
Modelling and Analysis, Proceeding of the 10th International Conference MMA2005 & CMAM2, 467-472.
[12] J.R. McMahon. Knot selection for least squares approximation using thin plate splines. M.S. Thesis, Naval Postgraduate School, 1986.
[13] J.R. McMahon, R. Franke. Knot selection for least squares thin plate splines. Technical Report, Naval Postgraduate School, Monterey, 1987.
[14] R.J. Renka, A.K. Cline. A triangle-based C1 interpolation method. Rocky Mountain J. Math., 14:223–237, 1984.
[15] R.J. Renka. Multivariate interpolation of large sets of scattered data. ACM Trans. Math. Software, 14:139–148, 1988.
[16] D. Shepard. A two dimensional interpolation function for irregularly spaced data. Proceedings of the 23rd ACM National Conference, New
York: ACM Press, 517–523, 1968.