A method of adding-removing knots for solving smoothing problems with obstacles. (English) Zbl 1179.65015
Summary: We study a method of adding-removing knots that has been proposed in the literature for solving the smoothing problem with obstacles. The method uses the coefficients of natural splines in the expansion by radial basis functions. We present examples of cycling and counterexamples to possible use of some ideas. We also give some sufficient conditions for finiteness of the method.
MSC:
| 65D07 | Numerical computation using splines |
References:
| [1] | Asmuss, S.; Budkina, N.; Oja, P., On smoothing problems with weights and obstacles, Proc. Estonian Acad. Sci. Phys. Math., 46, 262-272 (1997) · Zbl 0905.41006 |
| [2] | Budkina, N., On a method of solving a smoothing problem, Proc. Estonian Acad. Sci. Phys. Math., 45, 178-186 (1996) · Zbl 0912.65006 |
| [3] | Budkina, N., On the construction of smoothing splines by quadratic programming, Proc. Estonian Acad. Sci. Phys. Math., 49, 21-27 (2000) · Zbl 0971.65010 |
| [4] | Buhmann, M. D., Radial basis functions: theory and implementations. Radial basis functions: theory and implementations, Cambridge Monographs on Applied and Computational Mathematics (2003), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 1038.41001 |
| [5] | Buhmann, M. D.; Le Méhauté, A., Knot removal with radial function interpolation, C.R. Acad. Sci. Paris Sér. I Math., 320, 501-506 (1995) · Zbl 0839.65003 |
| [6] | Daugavet, V. A., Modification of the Wolfe method, Zh. Vychisl. Mat. i Mat. Fiz., 21, 504-508 (1981), (in Russian) · Zbl 0468.90068 |
| [7] | Ignatov, M. I.; Pevnyj, A. B., Reinsch’s method for solving the smoothing problem in the multidimensional case, Zh. Vychisl. Mat. i Mat. Fiz., 30, 186-192 (1990), (in Russian) · Zbl 0696.65009 |
| [8] | Ignatov, M. I.; Pevnyj, A. B., Natural Splines of Several Variables (1991), Nauka, Leningrad. Otdel.: Nauka, Leningrad. Otdel. Leningrad, (in Russian) · Zbl 0732.41012 |
| [9] | Kersey, S. N., On the problems of smoothing and near-interpolation, Math. Comp., 72, 1873-1885 (2003) · Zbl 1044.41002 |
| [10] | Kersey, S. N., Near-interpolation, Numer. Math., 94, 523-540 (2003) · Zbl 1107.41002 |
| [11] | Reinsch, C. H., Smoothing by spline functions, Numer. Math., 10, 177-183 (1967) · Zbl 0161.36203 |
| [12] | Reinsch, C. H., Smoothing by spline functions II, Numer. Math., 16, 451-454 (1971) · Zbl 1248.65020 |
| [13] | Wendland, H., Scattered data approximation, Cambridge Monographs on Applied and Computational Mathematics (2005), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 1075.65021 |
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