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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2014, Issue 1, Pages 141–152 (Mi vuu423)

COMPUTER SCIENCE

Exact solution of optimization task generated by simplest wave equation

N. V. Rodionova

Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia

Abstract: In the previous paper of the author the parameter family of finite-dimensional spaces of special quadratic splines of Lagrange's type has been defined. In each space, as a solution to the initial-boundary problem for the simplest wave equation, we have proposed the optimal spline, which gives the smallest residual. We have obtained exact formulas for coefficients of this spline and its residual. The formula for coefficients of this spline is a linear form of initial finite differences. The formula for the residual is a positive definite quadratic form of these quantities, but because of its bulkiness it is ill-suited for analyzing of the approximation quality of the input problem at the variation with the parameters.
For the purposes of the present paper, we have obtained an alternative representation for the residual, which is the positive definite quadratic form of the new finite differences defined on the boundary. The elements of the matrix of form are expressed in terms of Chebyshev's polynomials, the matrix is invertible and the inverse matrix has a tridiagonal form. This feature allows us to obtain, for the spectrum of the matrix, upper and lower bounds that are independent of the dimension $N$. Said fact allows us to make a study of the quality of approximation for different dimensions $N$ and weights $\omega\in[-1,1]$. It is shown that the parameter $\omega=0$ gives the best approximation and the residual tends to zero as $N$ increasing.

Keywords: interpolation, approximate spline, Chebyshev's polynomials.

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UDC: 519.651+517.518.823
MSC: 41A15

Citation: N. V. Rodionova, “Exact solution of optimization task generated by simplest wave equation”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2014, no. 1, 141–152

Citation in format AMSBIB
\Bibitem{Rod14} \by N.~V.~Rodionova \paper Exact solution of optimization task generated by simplest wave equation \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2014 \issue 1 \pages 141--152 \mathnet{http://mi.mathnet.ru/vuu423} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. I. Rodionov, “O reshenii odnoi zadachi optimizatsii, porozhdennoi prosteishim uravneniem teploprovodnosti”, Izv. IMI UdGU, 2014, no. 1(43), 49–67
2. V. I. Rodionov, “O lineinom algoritme chislennogo resheniya kraevoi zadachi dlya prosteishego volnovogo uravneniya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 25:1 (2015), 126–144
3. A. N. Mzedavee, V. I. Rodionov, “Tochnoe reshenie odnoi zadachi optimizatsii, porozhdennoi trekhmernym uravneniem Laplasa”, Izv. IMI UdGU, 51 (2018), 52–78
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