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 Tr. Inst. Mat., 2011, Volume 19, Number 1, Pages 32–44 (Mi timb137)

A principle for study of quasi-gradient methods of approximate solving operator equations in Hilbert spaces

O. N. Evkhutaa, P. P. Zabreikob

a South-Russia State Technical University
b Belarusian State University

Abstract: The article deals with nonlinear operator equations $f(x)=0$ with operators $f$ defined on a ball $B(x_0,R)$ in a Hilbert space $X$ and taking values from $X$. It is considered iterative methods of type $x_{n+1}=x_n-\Lambda(x_n)T(x_n)$, $n=0,1,2,ldots$, where $T(\xi)$ is an operator from $B(x_0,R)$ into $X$ and $\Lambda(\xi)$ a real functional on on $B(x_0,R)$. It is described conditions under that there is a phenomenon of relaxation of residuals: $\|f(x_{n+1}\|<\|f(x_n)\|$. The study of the convergence of iterations and its rate us reduce to the analysis of a scalar function; the graph of this function determines as the conditions of the convergence of iterations well as the rate of this convergence; moreover, it allows to write simple a priori and a posteriori estimates of errors. The general scheme covers classical methods of minimal residuals, of steepest descent, of minimal errors and some others.

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Document Type: Article
UDC: 513.88

Citation: O. N. Evkhuta, P. P. Zabreiko, “A principle for study of quasi-gradient methods of approximate solving operator equations in Hilbert spaces”, Tr. Inst. Mat., 19:1 (2011), 32–44

Citation in format AMSBIB
\Bibitem{EvkZab11} \by O.~N.~Evkhuta, P.~P.~Zabreiko \paper A principle for study of quasi-gradient methods of approximate solving operator equations in Hilbert spaces \jour Tr. Inst. Mat. \yr 2011 \vol 19 \issue 1 \pages 32--44 \mathnet{http://mi.mathnet.ru/timb137}