A method is proposed for the solution of some operator equations of the first kind by successive smoothing of the righthand side using an explicit spline approximation and by numerical inversion of the operator. For the relevant class of problems, the regularizing properties of the method are reduced on the whole to the regularizing properties of explicit spline approximation, which are studied under certain assumptions about errors in input data and the discretization step.
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A. N. Tikhonov, "On solution of ill-posed problems and regularization method,"Dokl. Akad. Nauk SSSR 151, No. 3, 501–504 (1963).
A. N. Tikhonov and V. Ya. Arsenin,Methods of Solution of Ill-Posed Problems [in Russian], Nauka, Moscow (1979).
A. N. Tikhonov, A. V. Goncharskii, V. V. Stepanov, and A. G. Yagola,Regularizing Algorithms and Prior Information [in Russian], Nauka, Moscow (1983).
A. I. Grebennikov, "Regularizing algorithms for solving some ill-posed problems by splines," in:Methods and Algorithms in Numerical Analysis [in Russian], Moscow State Univ. (1984).
C. H. Reinsch, "Smoothing by spline functions,"Num. Math. 10, 177–187 (1967).
V. A. Morozov,Regular Methods of Solution of Ill-Posed Problems [in Russian], Moscow State Univ. (1974).
V. A. Vasilenko,Spline Functions: Theory, Algorithms, Programs [in Russian], Nauka, Novosibirsk (1983).
A. I. Grebennikov,Spline Method for Solving Ill-Posed Problems of Approximation Theory [in Russian], Moscow State Univ. (1983).
I. S. Berezin and N. P. Zhidkov,Computational Methods [in Russian], Vol. 1, Nauka, Moscow (1966).
Yu. S. Zav'yalov, B. I. Kvasov, and V. L. Miroshnichenko,Spline Function Methods [in Russian], Nauka, Moscow (1980).
A. I. Grebennikov, "Spline collocation and double spline approximation methods for solving operator equations and application to the solution of integral equations with singularities," in:Methods and Algorithms in Numerical Analysis [in Russian], Moscow State Univ. (1984).
Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 39–46, 1986.
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Grebennikov, A.I. Regularizing properties of explicit approximating splines. Comput Math Model 1, 132–136 (1990). https://doi.org/10.1007/BF01129054
- Mathematical Modeling
- Input Data
- Computational Mathematic
- Industrial Mathematic
- Operator Equation