Abstract
We obtain some new Shepard type operators based on the classical, the modified Shepard methods and the least squares thin plate spline function. Given some sets of points, we compute some representative subsets of knot points following an algorithm described by J. R. McMahon in 1986.
Authors
Teodora Catinas
Babes-Bolyai University, Faculty of Mathematics and Computer Sciences
Babes-Bolyai University, Faculty of Mathematics and Computer Sciences
Babes-Bolyai University, Faculty of Mathematics and Computer Sciences
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Keywords
Scattered data; Shepard operator; least squares approximation; thinplate spline; knot points.
Paper coordinates
Malina Andra, Catinas Teodora, Shepard operator of least squares thin-plate spline type, Stud. Univ. Babes-Bolyai Math. 66(2021), No. 2, 257–265
http://doi.org/10.24193/subbmath.2021.2.02
About this paper
Print ISSN
0252-1938
Online ISSN
2065-961X
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Google Scholar
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[2] Catinas, T., 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-Birkhauser Verlag, 151(2005), 77-89.
[3] Catinas, T., Bivariate interpolation by combined Shepard operators, Proceedings of 17th IMACS World Congress, Scientific Computation, Applied Mathematics and Simulation, ISBN 2-915913-02-1, 2005, 7 pp.
[4] Catinas, T., The bivariate Shepard operator of Bernoulli type, Calcolo, 44 (2007), no. 4, 189-202.
[5] Coman, Gh., The remainder of certain Shepard type interpolation formulas, Stud. Univ. Babes-Bolyai Math., 32(1987), no. 4, 24-32.
[6] Coman, Gh., Hermite-type Shepard operators, Rev. Anal. Num´er. Th´eor. Approx.,
26(1997), 33-38.
[7] Coman, Gh., Shepard operators of Birkhoff type, Calcolo, 35(1998), 197-203.
[8] Farwig, R., Rate of convergence of Shepard’s global interpolation formula, Math. Comp., 46(1986), 577-590.
[9] Franke, R., Scattered data interpolation: tests of some methods, Math. Comp., 38(1982), 181-200.
[10] Franke, R., Nielson, G., Smooth interpolation of large sets of scattered data, Int. J. Numer. Meths. Engrg., 15(1980), 1691-1704.
[11] Lazzaro, D., Montefusco, L.B., Radial basis functions for multivariate interpolation of large scattered data sets, J. Comput. Appl. Math., 140(2002), 521-536.
[12] McMahon, J.R., Knot selection for least squares approximation using thin plate splines, M.S. Thesis, Naval Postgraduate School, 1986.
[13] McMahon, J.R., Franke, R., Knot selection for least squares thin plate splines, Technical Report, Naval Postgraduate School, Monterey, 1987.
[14] Renka, R.J., Multivariate interpolation of large sets of scattered data, ACM Trans. Math. Software, 14(1988), 139-148.
[15] Renka, R.J., Cline, A.K., A triangle-based C1 interpolation method, Rocky Mountain J. Math., 14(1984), 223-237.
[16] Shepard, D., A two dimensional interpolation function for irregularly spaced data, Proc. 23rd Nat. Conf. ACM, 1968, 517-523.
[17] Trımbitas, G., Combined Shepard-least squares operators – computing them using spatial data structures, Stud. Univ. Babes-Bolyai Math., 47(2002), 119-128.
[18] Zuppa, C., Error estimates for moving least square approximations, Bull. Braz. Math. Soc., New Series, 34(2003), no. 2, 231-249.
[2] Catinas, T., 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-Birkhauser Verlag, 151(2005), 77-89.
[3] Catinas, T., Bivariate interpolation by combined Shepard operators, Proceedings of 17th IMACS World Congress, Scientific Computation, Applied Mathematics and Simulation, ISBN 2-915913-02-1, 2005, 7 pp.
[4] Catinas, T., The bivariate Shepard operator of Bernoulli type, Calcolo, 44 (2007), no. 4, 189-202.
[5] Coman, Gh., The remainder of certain Shepard type interpolation formulas, Stud. Univ. Babes-Bolyai Math., 32(1987), no. 4, 24-32.
[6] Coman, Gh., Hermite-type Shepard operators, Rev. Anal. Num´er. Th´eor. Approx.,
26(1997), 33-38.
[7] Coman, Gh., Shepard operators of Birkhoff type, Calcolo, 35(1998), 197-203.
[8] Farwig, R., Rate of convergence of Shepard’s global interpolation formula, Math. Comp., 46(1986), 577-590.
[9] Franke, R., Scattered data interpolation: tests of some methods, Math. Comp., 38(1982), 181-200.
[10] Franke, R., Nielson, G., Smooth interpolation of large sets of scattered data, Int. J. Numer. Meths. Engrg., 15(1980), 1691-1704.
[11] Lazzaro, D., Montefusco, L.B., Radial basis functions for multivariate interpolation of large scattered data sets, J. Comput. Appl. Math., 140(2002), 521-536.
[12] McMahon, J.R., Knot selection for least squares approximation using thin plate splines, M.S. Thesis, Naval Postgraduate School, 1986.
[13] McMahon, J.R., Franke, R., Knot selection for least squares thin plate splines, Technical Report, Naval Postgraduate School, Monterey, 1987.
[14] Renka, R.J., Multivariate interpolation of large sets of scattered data, ACM Trans. Math. Software, 14(1988), 139-148.
[15] Renka, R.J., Cline, A.K., A triangle-based C1 interpolation method, Rocky Mountain J. Math., 14(1984), 223-237.
[16] Shepard, D., A two dimensional interpolation function for irregularly spaced data, Proc. 23rd Nat. Conf. ACM, 1968, 517-523.
[17] Trımbitas, G., Combined Shepard-least squares operators – computing them using spatial data structures, Stud. Univ. Babes-Bolyai Math., 47(2002), 119-128.
[18] Zuppa, C., Error estimates for moving least square approximations, Bull. Braz. Math. Soc., New Series, 34(2003), no. 2, 231-249.
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