Cubic convergence order yielding iterative regularization methods for ill-posed Hammerstein type operator equations

2016 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George ◽  
Shobha Monnanda Erappa
Keyword(s):  
Type Operator ◽  
Convergence Order ◽  
Regularization Methods ◽  
Iterative Regularization ◽  
Operator Equations ◽  
Cubic Convergence ◽  
Ill Posed

scholarly journals On fractional asymptotical regularization of linear ill-posed problems in hilbert spaces

2019 ◽  
Vol 22 (3) ◽  
pp. 699-721 ◽  
Author(s):  
Ye Zhang ◽  
Bernd Hofmann
Keyword(s):  
Fractional Order ◽  
Hilbert Spaces ◽  
Regularization Method ◽  
Numerical Realization ◽  
Regularization Methods ◽  
Iterative Regularization ◽  
Operator Equations ◽  
Ill Posed ◽  
One Step ◽  
The One

Abstract In this paper, we study a fractional-order variant of the asymptotical regularization method, called Fractional Asymptotical Regularization (FAR), for solving linear ill-posed operator equations in a Hilbert space setting. We assign the method to the general linear regularization schema and prove that under certain smoothness assumptions, FAR with fractional order in the range (1, 2) yields an acceleration with respect to comparable order optimal regularization methods. Based on the one-step Adams-Moulton method, a novel iterative regularization scheme is developed for the numerical realization of FAR. Two numerical examples are given to show the accuracy and the acceleration effect of FAR.

Accuracy estimates of Gauss–Newton-type iterative regularization methods for nonlinear equations with operators having normally solvable derivative at the solution

2016 ◽  
Vol 24 (4) ◽  
Author(s):  
Mikhail Y. Kokurin
Keyword(s):  
Hilbert Spaces ◽  
Total Error ◽  
A Priori ◽  
Stopping Rules ◽  
Regularization Methods ◽  
Iterative Regularization ◽  
Operator Equations ◽  
Ill Posed ◽  
Irregular Operator ◽  
The Right

AbstractWe investigate a class of iterative regularization methods for solving nonlinear irregular operator equations in Hilbert spaces. The operator of an equation is supposed to have a normally solvable derivative at the desired solution. The operators and right parts of equations can be given with errors. A priori and a posteriori stopping rules for the iterations are analyzed. We prove that the accuracy of delivered approximations is proportional to the total error level in the operator and the right part of an equation. The obtained results improve known accuracy estimates for the class of iterative regularization methods, as applied to general irregular operator equations. The results also extend previous similar estimates related to regularization methods for linear ill-posed equations with normally solvable operators.

Iterative Regularization Methods for Nonlinear Ill-Posed Problems

2008 ◽  
Author(s):  
Barbara Kaltenbacher ◽  
Andreas Neubauer ◽  
Otmar Scherzer
Keyword(s):  
Regularization Methods ◽  
Iterative Regularization ◽  
Ill Posed

scholarly journals A two step newton type iteration for ill-posed Hammerstein type operator equations in Hilbert scales

2014 ◽  
Vol 26 (1) ◽  
pp. 91-116
Author(s):  
Monnanda Erappa Shobha ◽  
Santhosh George ◽  
M. Kunhanandan
Keyword(s):  
Type Operator ◽  
Operator Equations ◽  
Ill Posed

Convergence Rates For Lavrentiev-Type Regularization In Hilbert Scales

2008 ◽  
Vol 8 (3) ◽  
pp. 279-293 ◽  
Author(s):  
M.T. NAIR ◽  
U. TAUTENHAHN
Keyword(s):  
Error Bounds ◽  
Convergence Rates ◽  
Noisy Data ◽  
Regularization Methods ◽  
Operator Equations ◽  
Optimal Error ◽  
Regularization Scheme ◽  
Data Regularization ◽  
Ill Posed ◽  
Adjoint Operators

AbstractFor solving linear ill-posed problems with noisy data regularization methods are required. We analyze a simplified regularization scheme in Hilbert scales for operator equations with nonnegative self-adjoint operators. By exploiting the op-erator monotonicity of certain functions, order-optimal error bounds are derived that characterize the accuracy of the regularized approximations. These error bounds have been obtained under general smoothness conditions.

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