R-K type Landweber method for nonlinear ill-posed problems
type: Item
instance of: scholarly article
title: R-K type Landweber method for nonlinear ill-posed problems
zbMATH Open document ID: 1131.65047
author: Li Li
author: Bo Han
author: Wei Wang
published in: Journal of Computational and Applied Mathematics
publication date: 2007-06-29T00:00:00Z
review text: The paper concerns a regularization method for solving a nonlinear ill-posed problem \(F(x)=y\) in a Hilbert space. The Runge-Kutta (R-K) method of order 2 is applied to the continuous analog of the Landweber method: \(\dot{x}^\delta(t)=F'(x^\delta(t))^*[y^\delta-F(x^\delta(t))]\), \(0<t\leq T\), \(x^\delta(0)=x_0\). Here \(y^\delta\) are available noisy data with \(\| y-y^\delta\| \leq\delta\). This method, called by the authors the Runge-Kutta type Landweber method, gives a sequence of regularized solutions \(\{x^\delta_k\}\), where \(k\) plays the role of regularization parameter. Applying a generalized discrepancy principle as the stopping rule (\(k\leq k_*(\delta)\)) the authors prove the convergence when \(\delta\rightarrow 0\). Moreover, a convergence rate is obtained under a source condition and usual additional assumptions on \(F\). Finally, numerical results of the considered method applied to the convolution equation are compared with the corresponding results for the Landweber method.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 47J06
zbMATH DE Number: 5168509
zbMATH Keywords: nonlinear ill-posed problems
zbMATH Keywords: Landweber iterations
zbMATH Keywords: continuous Landweber method
zbMATH Keywords: Runge-Kutta method
zbMATH Keywords: convergence
zbMATH Keywords: generalized discrepancy principle
zbMATH Keywords: stability estimate
zbMATH Keywords: Hilbert space
zbMATH Keywords: regularization
zbMATH Keywords: numerical results
reviewed by: Teresa Regińska
MaRDI profile type: Publication
full work available at URL: j.cam.2006.07.022
OpenAlex ID: W1966907405
cites work: Q4517809
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DOI: 10.1016/J.CAM.2006.07.022
description: scientific article