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SPRING 2014 A two step newton type iteration for ill-posed Hammerstein type operator equations in Hilbert scales
Monnanda Erappa Shobha, Santhosh George, M. Kunhanandan
J. Integral Equations Applications 26(1): 91-116 (SPRING 2014). DOI: 10.1216/JIE-2014-26-1-91

Abstract

In this paper regularized solutions of ill-posed Hammerstein type operator equation $KF(x)=y$, where $K:X\rightarrow Y$ is a bounded linear operator with non-closed range and $F:X\rightarrow X$ is non-linear, are obtained by a two step Newton type iterative method in Hilbert scales, where the available data is $y^{\delta}$ in place of actual data $y$ with $\|y-y^\delta\|\leq\delta$. We require only a weaker assumption $\|F^{\p}(x_0)x\|\sim\|x\|_{-b}$ compared to the usual assumption $\|F^{\p}(\widehat{x})x\|\sim\|x\|_{-b}$, where $\widehat{x}$ is the actual solution of the problem, which is assumed to exist, and $x_0$ is the initial approximation. Two cases, viz-a-viz, (i)~when $F^{\p}(x_0)$ is boundedly invertible and (ii)~$F^{\p}(x_0)$ is non-invertible but $F$ is monotone operator, are considered. We derive error bounds under certain general source conditions by choosing the regularization parameter by an a~priori manner as well as by using a modified form of the adaptive scheme proposed by Perverzev and Schock [{\bf14}].

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Monnanda Erappa Shobha. Santhosh George. M. Kunhanandan. "A two step newton type iteration for ill-posed Hammerstein type operator equations in Hilbert scales." J. Integral Equations Applications 26 (1) 91 - 116, SPRING 2014. https://doi.org/10.1216/JIE-2014-26-1-91

Information

Published: SPRING 2014
First available in Project Euclid: 17 April 2014

zbMATH: 1288.65083
MathSciNet: MR3195116
Digital Object Identifier: 10.1216/JIE-2014-26-1-91

Subjects:
Primary: 47A50 , 47H17 , 65J15 , 65J20

Keywords: adaptive choice , Hilbert scales , Ill-posed problems , Newton's method , Tikhonov regularization

Rights: Copyright © 2014 Rocky Mountain Mathematics Consortium

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Vol.26 • No. 1 • SPRING 2014
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