Convergence rates for the iteratively regularized Gauss–Newton method in Banach spaces

2010 ◽  
Vol 26 (3) ◽  
pp. 035007 ◽  
Author(s):  
Barbara Kaltenbacher ◽  
Bernd Hofmann
Keyword(s):  
Banach Spaces ◽  
Newton Method ◽  
Convergence Rates

scholarly journals The Ivanov regularized Gauss–Newton method in Banach space with an a posteriori choice of the regularization radius

2019 ◽  
Vol 27 (4) ◽  
pp. 539-557
Author(s):  
Barbara Kaltenbacher ◽  
Andrej Klassen ◽  
Mario Luiz Previatti de Souza
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Newton Method ◽  
Noise Level ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
A Priori ◽  
Discrepancy Principle ◽  
A Posteriori ◽  
Source Condition

Abstract In this paper, we consider the iteratively regularized Gauss–Newton method, where regularization is achieved by Ivanov regularization, i.e., by imposing a priori constraints on the solution. We propose an a posteriori choice of the regularization radius, based on an inexact Newton/discrepancy principle approach, prove convergence and convergence rates under a variational source condition as the noise level tends to zero and provide an analysis of the discretization error. Our results are valid in general, possibly nonreflexive Banach spaces, including, e.g., {L^{\infty}} as a preimage space. The theoretical findings are illustrated by numerical experiments.

On smoothness conditions and convergence rates in the CLT in Banach spaces

1993 ◽  
Vol 96 (2) ◽  
pp. 137-151
Author(s):  
Vidmantas Bentkus ◽  
Friedrich G�tze
Keyword(s):  
Banach Spaces ◽  
Convergence Rates

On convergence rates for iteratively regularized Newton-type methods under a Lipschitz-type nonlinearity condition

2015 ◽  
Vol 23 (1) ◽  
Author(s):  
Frank Werner
Keyword(s):  
Newton Method ◽  
Convergence Rates ◽  
Poisson Noise ◽  
Type Condition ◽  
General Data ◽  
Data Fidelity ◽  
Forward Operator ◽  
General Error

AbstractWe investigate a generalization of the well-known iteratively regularized Gauss–Newton method where the Newton equations are regularized variationally using general data fidelity and penalty terms. To obtain convergence rates, we use a general error assumption which has recently been shown to be useful for impulsive and Poisson noise. We restrict the nonlinearity of the forward operator only by a Lipschitz-type condition and compare our results to other convergence rates results proven in the literature. Finally we explicitly state our convergence rates for the aforementioned case of Poisson noise to shed some light on the structure of the posed error assumption.

scholarly journals A Banach Space Regularization Approach for Multifrequency Microwave Imaging

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Claudio Estatico ◽  
Alessandro Fedeli ◽  
Matteo Pastorino ◽  
Andrea Randazzo
Keyword(s):  
Banach Space ◽  
Data Processing ◽  
Numerical Simulations ◽  
Banach Spaces ◽  
Newton Method ◽  
Mathematical Formulation ◽  
Processing Technique ◽  
Microwave Imaging ◽  
Inexact Newton Method ◽  
Reconstruction Method

A method for microwave imaging of dielectric targets is proposed. It is based on a tomographic approach in which the field scattered by an unknown target (and collected in a proper observation domain) is inverted by using an inexact-Newton method developed inLpBanach spaces. In particular, the extension of the approach to multifrequency data processing is reported. The mathematical formulation of the new method is described and the results of numerical simulations are reported and discussed, analyzing the behavior of the multifrequency processing technique combined with the Banach spaces reconstruction method.

Error estimates for the simplified iteratively regularized Gauss–Newton method in Banach spaces under a Morozov-type stopping rule

2018 ◽  
Vol 26 (3) ◽  
pp. 311-333 ◽  
Author(s):  
Pallavi Mahale ◽  
Sharad Kumar Dixit
Keyword(s):  
Banach Spaces ◽  
Error Estimates ◽  
Newton Method ◽  
A Priori ◽  
Identification Problem ◽  
Stopping Rule ◽  
Optimal Error Estimates ◽  
Optimal Error ◽  
Parameter Identification Problem ◽  
Ill Posed

AbstractJin Qinian and Min Zhong [10] considered an iteratively regularized Gauss–Newton method in Banach spaces to find a stable approximate solution of the nonlinear ill-posed operator equation. They have considered a Morozov-type stopping rule (Rule 1) as one of the criterion to stop the iterations and studied the convergence analysis of the method. However, no error estimates have been obtained for this case. In this paper, we consider a modified variant of the method, namely, the simplified Gauss–Newton method under both an a priori as well as a Morozov-type stopping rule. In both cases, we obtain order optimal error estimates under Hölder-type approximate source conditions. An example of a parameter identification problem for which the method can be implemented is discussed in the paper.

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