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.

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).

Derivative Free Regularization Method for Nonlinear Ill-Posed Equations in Hilbert Scales

2019 ◽  
Vol 19 (4) ◽  
pp. 765-778
Author(s):  
Santhosh George ◽  
K. Kanagaraj
Keyword(s):  
Regularization Parameter ◽  
Adjoint Operator ◽  
Identification Problem ◽  
Optimal Order ◽  
Order Error ◽  
Adaptive Parameter ◽  
Derivative Free ◽  
Parameter Identification Problem ◽  
Ill Posed ◽  
Parameter Choice Strategy

AbstractIn this paper, we deal with nonlinear ill-posed operator equations involving a monotone operator in the setting of Hilbert scales. Our convergence analysis of the proposed derivative-free method is based on the simple property of the norm of a self-adjoint operator. Using a general Hölder-type source condition, we obtain an optimal order error estimate. Also we consider the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter. Finally, we applied the proposed method to the parameter identification problem in an elliptic PDE in the setting of Hilbert scales and compare the results with the corresponding method in Hilbert space.

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 ITERATED LAVRENTIEV REGULARIZATION FOR NONLINEAR ILL-POSED PROBLEMS

2009 ◽  
Vol 51 (2) ◽  
pp. 191-217 ◽  
Author(s):  
P. MAHALE ◽  
M. T. NAIR
Keyword(s):  
Nonlinear Operator ◽  
Optimal Error Estimate ◽  
Positive Real ◽  
Iterative Regularization ◽  
Optimal Error ◽  
Lavrentiev Regularization ◽  
Inexact Data ◽  
Ill Posed ◽  
Funct Anal ◽  
General Source

AbstractWe consider an iterated form of Lavrentiev regularization, using a null sequence (αk) of positive real numbers to obtain a stable approximate solution for ill-posed nonlinear equations of the form F(x)=y, where F:D(F)⊆X→X is a nonlinear operator and X is a Hilbert space. Recently, Bakushinsky and Smirnova [“Iterative regularization and generalized discrepancy principle for monotone operator equations”, Numer. Funct. Anal. Optim.28 (2007) 13–25] considered an a posteriori strategy to find a stopping index kδ corresponding to inexact data yδ with $\|y-y^\d \|\leq \d $ resulting in the convergence of the method as δ→0. However, they provided no error estimates. We consider an alternate strategy to find a stopping index which not only leads to the convergence of the method, but also provides an order optimal error estimate under a general source condition. Moreover, the condition that we impose on (αk) is weaker than that considered by Bakushinsky and Smirnova.

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.

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