UDC 519.6:004.942
1 G.E. Pukhov Institute for Modelling in Energy Engineering, National Academy of Sciences of Ukraine, Kyiv, Ukraine
 afverl@gmail.com
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2 Ya.S. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine
Petro.Malachivskyy@gmail.com
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SOLVING THE PROBLEM OF INTERPRETING OBSERVATIONS USING
THE SPLINE APPROXIMATION OF THE SCANNED FUNCTION
Abstract. An accuracy analysis of the numerical implementation of the frequency method for solving the integral equation
in the problem of interpreting technical observations using the spline approximation of the scanned function is presented.
The algorithm for solving the integral equation of the interpretation problem, which is based on the application
of the Tikhonov regularization method with the search for a solution in the frequency domain with a truncation
of the frequency spectrum is investigated. To increase the accuracy of the interpretation results, the use of spline approximation
of the values of the scanned function, i.e., the right-hand side of the integral equation, is proposed. An estimate of the accuracy
of solving the integral equation using the regularization method and taking into account the error accompanied by the inaccuracy
of the right-hand side, as well as the error in calculating the values of the kernel is obtained. A method for calculating the optimal degree
of smoothing spline for approximation of the scanned function that provides the required accuracy is proposed.
Keywords: interpretation problem, Fredholm integral equation, Tikhonov regularization method, frequency spectrum truncation,
spline approximation, accuracy estimation.
FULL TEXT
REFERENCES
- Verlan A.F., Sizikov V.S. Integral equations: methods, algorithms, programs. Reference manual [in Russian]. Kiev: Nauk. dumka, 1986. 542 p.
- Verlan A.F., Goroshko I.O., Karpenko E.Yu., Korolev V.Yu., Mosentsova L.V. Methods and algorithms for recovering signals and images[in Russian]. Kiev: NAS of Ukraine, Institute of Modeling Problems in Power Engineering. G.E. Pukhova, 2011. 368 p.
- Starkov V.N. Constructive methods of computational physics in interpretation problems [in Russian]. Kiev: Nauk. dumka, 2002. 264 p.
- Godlevsky V.S., Godlevsky V.V. Signal processing accuracy issues [in Russian]. Kiev: Alpha Reklama, 2020. 407 p.
- Sergienko I.V., Zadiraka V.K., Lytvyn O.M., Melnikova S.S., Nechuyviter O.P. Optimal algorithms for calculating integrals from fast-oscillating functions and their application. Vol. 1. Algorithms. 447 p., Vol. 2. Applications. 348 p. Kyiv: Nauk. opinion, 2011.
- Sergienko I.V., Zadiraka V.K., Lytvyn O.M. Elements of the general theory of optimal algorithms and related issues [in Ukrainian]. Kyiv: Nauk. dumka, 2012. 400 p.
- Khimich O.M., Nikolaevskaya O.A., Chistyakova T.V. About some ways to improve the accuracy of computer calculations. Mathematical and computer modeling. Series: Physical and Mathematical Sciences. Coll. Science. works. V.M.Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine, Kamyanets-Podilsky National. I. Ogienko University, 2017, Iss. 15. P. 249–254.
- Starkov V.N., Borshch A.A., Gandzha I.S., Tomchuk P.M. Some examples of seemingly plausible interpretation of experimental results. Ukr. J. Phys. 2017. Vol. 62, N 6. P. 481–488.
- Verlan D.A. The method of degenerate kernels in the numerical implementation of integral dynamic models. Elektronnoye modelirovaniye. 2014. Vol. 36, N 3. P. 41–58.
- Verlan D.A., Ponedilok V.V. Numerical implementation of integrated dynamic models based on the method of degenerate core. Mathematical and computer modeling. Series: Technical Sciences. 2019. Iss. 20. P. 131–145.
- Verlan A.F., Fedorchuk V.A. Signal recovery in surveillance and control systems based on the solution of the inverse problem with truncation of the spectrum of the core of the integrated operator. Mathematical and computer modeling. Series: Technical Sciences. 2018. Iss. 17. P. 5–15.
- Verlan A.F., Verlan A.A., Polozhaenko S.A. Algorithmization of methods of precision parametric reduction of mathematical models. Informatics and mathematical methods in modeling. 2017. Vol. 7, N 1–2. P. 7–18.
- Collatz L., Krabs W. Approximationstheorie: Tschebyscheffsche Approximation mit Anwendungen (Teubner Studienbcher Mathematik). Stuttgart: Teubner Verlag; 1973.
- Malachivsky P.S., Skopetsky V.V. Continuous and smooth minimax spline approximation [in Ukrainian]. Kyiv: Nauk. dumka, 2013. 270 p.
- Malachivskyy P.S., Pizyur Y.V., Malachivskyi R.P., Ukhanska O.M. Chebyshev approximation of functions of several variables. Cybernetics and Systems Analysis. 2020 Vol. 56, N 1. Р. 118–125. http://doi.org/10.1007/s10559-020-00227-8.
- Zavyalov Yu.S., Kvasov B.I., Miroshnichenko V.L. Spline function methods [in Russian]. Moscow: Nauka, 1980. 350 p.
- Grebennikov A.I. The spline method and the solution of ill-posed problems in approximation theory [in Russian]. Moscow: Mosk. un-t, 1983. 208 p.
- Berdyshev V.I., Subbotin Yu.N. Numerical methods for approximating functions [in Russian]. Sverdlovsk: Central Ural publishing, 1979. 120 p.
- Verlan D.A., Chevskaya K.S. Estimation of errors in solving Volterra integral equations of the second type by means of integral inequalities. Mathematical and computer modeling. Series: Technical Sciences. 2013. Iss. 9. P. 23–33.
