A UNIFIED APPROACH FOR REGULARIZING DISCRETIZED LINEAR ILL‐POSED PROBLEMS

2009 ◽  
Vol 14 (4) ◽  
pp. 451-466
Author(s):  
Torsten Hein
Keyword(s):  
Hilbert Spaces ◽  
Convergence Rates ◽  
Regularization Parameter ◽  
Unified Approach ◽  
Choice Strategy ◽  
A Posteriori Parameter Choice ◽  
Ill Posed ◽  
Parameter Choice Strategy ◽  
General Source ◽  
Unknown Solution

In this paper we deal with regularization approaches for discretized linear ill‐posed problems in Hilbert spaces. As opposite to other contributions concerning this topic the smoothness of the unknown solution is measured with so‐called approximative source conditions. This idea allows us to generalize known convergence rates results to arbitrary classes of smoothness conditions including logarithmic and general source conditions. The considerations include an a‐posteriori parameter choice strategy for the regularization parameter and the discretization level. Results of one numerical example are presented.

scholarly journals An optimal order yielding discrepancy principle for simplified regularization of ill-posed problems in Hilbert scales

2003 ◽  
Vol 2003 (39) ◽  
pp. 2487-2499 ◽  
Author(s):  
Santhosh George ◽  
M. Thamban Nair
Keyword(s):  
Approximate Solution ◽  
Hilbert Spaces ◽  
Regularization Parameter ◽  
Optimal Order ◽  
Discrepancy Principle ◽  
Parameter Choice ◽  
Bounded Linear ◽  
Choice Strategy ◽  
Ill Posed ◽  
Parameter Choice Strategy

Recently, Tautenhahn and Hämarik (1999) have considered a monotone rule as a parameter choice strategy for choosing the regularization parameter while considering approximate solution of an ill-posed operator equationTx=y, whereTis a bounded linear operator between Hilbert spaces. Motivated by this, we propose a new discrepancy principle for the simplified regularization, in the setting of Hilbert scales, whenTis a positive and selfadjoint operator. When the datayis known only approximately, our method provides optimal order under certain natural assumptions on the ill-posedness of the equation and smoothness of the solution. The result, in fact, improves an earlier work of the authors (1997).

Secant-type iteration for nonlinear ill-posed equations in Banach space

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Santhosh George ◽  
C. D. Sreedeep ◽  
Ioannis K. Argyros
Keyword(s):  
Iterative Scheme ◽  
Regularization Parameter ◽  
Optimal Error Estimate ◽  
Optimal Error ◽  
Adaptive Parameter ◽  
Choice Strategy ◽  
Accretive Mappings ◽  
Ill Posed ◽  
Adaptive Parameter Choice Strategy ◽  
Parameter Choice Strategy

Abstract In this paper, we study secant-type iteration for nonlinear ill-posed equations involving 𝑚-accretive mappings in Banach spaces. We prove that the proposed iterative scheme has a convergence order at least 2.20557 using assumptions only on the first Fréchet derivative of the operator. Further, using a general Hölder-type source condition, we obtain an optimal error estimate. We also use the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter.

A parameter choice strategy for a multilevel augmentation method in iterated Lavrentiev regularization

2018 ◽  
Vol 26 (2) ◽  
pp. 153-170 ◽  
Author(s):  
Chunmei Zeng ◽  
Xingjun Luo ◽  
Suhua Yang ◽  
Fanchun Li
Keyword(s):  
Integral Equation ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
Parameter Choice ◽  
Lavrentiev Regularization ◽  
Choice Strategy ◽  
Ill Posed ◽  
Parameter Choice Strategy

AbstractIn this paper we apply the multilevel augmentation method to solve an ill-posed integral equation via the iterated Lavrentiev regularization. This method leads to fast solutions of discrete iterated Lavrentiev regularization. The convergence rates of the iterated Lavrentiev regularization are achieved by using a certain parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method.

Multiscale Galerkin methods for the nonstationary iterated Tikhonov method with a modified posteriori parameter selection

2018 ◽  
Vol 26 (1) ◽  
pp. 109-120
Author(s):  
Xingjun Luo ◽  
Zhaofu Ouyang ◽  
Chunmei Zeng ◽  
Fanchun Li
Keyword(s):  
Convergence Rates ◽  
Regularization Parameter ◽  
Fredholm Integral Equations ◽  
Galerkin Methods ◽  
A Posteriori ◽  
Parameter Choice ◽  
Compression Technique ◽  
Choice Strategy ◽  
Optimal Convergence Rates ◽  
Parameter Choice Strategy

AbstractIn this paper, we consider a fast multiscale Galerkin method with compression technique for solving Fredholm integral equations of the first kind via the nonstationary iterated Tikhonov regularization. A modified a posteriori regularization parameter choice strategy is established, which leads to optimal convergence rates.

On the Parameter Choice in the Multilevel Augmentation Method

2020 ◽  
Vol 20 (3) ◽  
pp. 555-571
Author(s):  
Suhua Yang ◽  
Xingjun Luo ◽  
Chunmei Zeng ◽  
Zhihai Xu ◽  
Wenyu Hu
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Method ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
Tikhonov Regularization Method ◽  
Fredholm Integral Equations ◽  
Parameter Choice ◽  
Choice Strategy ◽  
Ill Posed ◽  
Parameter Choice Strategy

AbstractIn this paper, we apply the multilevel augmentation method for solving ill-posed Fredholm integral equations of the first kind via iterated Tikhonov regularization method. The method leads to fast solutions of the discrete regularization methods for the equations. The convergence rates of iterated Tikhonov regularization are achieved by using a modified parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method.

scholarly journals Newton-Type Iteration for Tikhonov Regularization of Nonlinear Ill-Posed Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Santhosh George
Keyword(s):  
Newton Method ◽  
Hilbert Spaces ◽  
Nonlinear Integral Equation ◽  
Nonlinear Operator ◽  
Regularization Parameter ◽  
Adaptive Method ◽  
Majorizing Sequence ◽  
Source Condition ◽  
Ill Posed ◽  
General Source

Recently in the work of George, 2010, we considered a modified Gauss-Newton method for approximate solution of a nonlinear ill-posed operator equationF(x)=y, whereF:D(F)⊆X→Yis a nonlinear operator between the Hilbert spacesXandY. The analysis in George, 2010 was carried out using a majorizing sequence. In this paper, we consider also the modified Gauss-Newton method, but the convergence analysis and the error estimate are obtained by analyzing the odd and even terms of the sequence separately. We use the adaptive method in the work of Pereverzev and Schock, 2005 for choosing the regularization parameter. The optimality of this method is proved under a general source condition. A numerical example of nonlinear integral equation shows the performance of this procedure.

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