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.

scholarly journals Two-Step Modified Newton Method for Nonlinear Lavrentiev Regularization

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Santhosh George ◽  
Suresan Pareth
Keyword(s):  
Newton Method ◽  
Regularization Parameter ◽  
Adaptive Method ◽  
Initial Guess ◽  
Optimal Order ◽  
Lavrentiev Regularization ◽  
Modified Newton Method ◽  
Actual Solution ◽  
Ill Posed ◽  
General Source

A two step modified Newton method is considered for obtaining an approximate solution for the nonlinear ill-posed equation F(x)=f when the available data are fδ with ‖f−fδ‖≤δ and the operator F is monotone. The derived error estimate under a general source condition on x0−x^ is of optimal order; here x0 is the initial guess and x^ is the actual solution. The regularization parameter is chosen according to the adaptive method considered by Perverzev and Schock (2005). The computational results provided endorse the reliability and effectiveness of our method.

Simplified Iteratively Regularized Gauss–Newton Method in Banach Spaces Under a General Source Condition

2020 ◽  
Vol 20 (2) ◽  
pp. 321-341
Author(s):  
Pallavi Mahale ◽  
Sharad Kumar Dixit
Keyword(s):  
Newton Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Identification Problem ◽  
Stopping Rule ◽  
Optimal Error Estimates ◽  
Source Condition ◽  
Parameter Identification Problem ◽  
Ill Posed ◽  
General Source

AbstractIn this paper, we consider a simplified iteratively regularized Gauss–Newton method in a Banach space setting under a general source condition. We will obtain order-optimal error estimates both for an a priori stopping rule and for a Morozov-type stopping rule together with a posteriori choice of the regularization parameter. An advantage of a general source condition is that it provides a unified setting for the error analysis which can be applied to the cases of both severely and mildly ill-posed problems. We will give a numerical example of a parameter identification problem to discuss the performance of the method.

Simplified Generalized Gauss–Newton Method for Nonlinear Ill-Posed Operator Equations in Hilbert Scales

2018 ◽  
Vol 18 (4) ◽  
pp. 687-702 ◽  
Author(s):  
Pallavi Mahale ◽  
Pradeep Kumar Dadsena
Keyword(s):  
Approximate Solution ◽  
Error Estimate ◽  
Newton Method ◽  
Operator Equation ◽  
Hilbert Spaces ◽  
Nonlinear Operator ◽  
Optimal Error Estimate ◽  
Operator Equations ◽  
Optimal Error ◽  
Ill Posed

AbstractIn this paper, we study the simplified generalized Gauss–Newton method in a Hilbert scale setting to get an approximate solution of the ill-posed operator equation of the form {F(x)=y} where {F:D(F)\subseteq X\to Y} is a nonlinear operator between Hilbert spaces X and Y. Under suitable nonlinearly conditions on F, we obtain an order optimal error estimate under the Morozov type stopping rule.

scholarly journals Inverse Free Iterative Methods for Nonlinear Ill-Posed Operator Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George ◽  
P. Jidesh
Keyword(s):  
Approximate Solution ◽  
Regularization Parameter ◽  
Optimal Order ◽  
Operator Equations ◽  
Source Condition ◽  
Numerical Examples ◽  
Ill Posed ◽  
Balancing Principle ◽  
General Source ◽  
New Iterative Method

We present a new iterative method which does not involve inversion of the operators for obtaining an approximate solution for the nonlinear ill-posed operator equationF(x)=y. The proposed method is a modified form of Tikhonov gradient (TIGRA) method considered by Ramlau (2003). The regularization parameter is chosen according to the balancing principle considered by Pereverzev and Schock (2005). The error estimate is derived under a general source condition and is of optimal order. Some numerical examples involving integral equations are also given in this paper.

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 Newton Type Iteration for Tikhonov Regularization of Nonlinear Ill-Posed Problems in Hilbert Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Monnanda Erappa Shobha ◽  
Santhosh George
Keyword(s):  
Integral Equation ◽  
Iterative Method ◽  
Iterative Scheme ◽  
Initial Guess ◽  
Optimal Order ◽  
Source Condition ◽  
Adaptive Scheme ◽  
Actual Solution ◽  
Ill Posed ◽  
General Source

Recently, Vasin and George (2013) considered an iterative scheme for approximately solving an ill-posed operator equationF(x)=y. In order to improve the error estimate available by Vasin and George (2013), in the present paper we extend the iterative method considered by Vasin and George (2013), in the setting of Hilbert scales. The error estimates obtained under a general source condition onx0-x^(x0is the initial guess andx^is the actual solution), using the adaptive scheme proposed by Pereverzev and Schock (2005), are of optimal order. The algorithm is applied to numerical solution of an integral equation in Numerical Example section.

scholarly journals Convergence Rate of the Modified Landweber Method for Solving Inverse Potential Problems

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 608
Author(s):  
Pornsarp Pornsawad ◽  
Parada Sungcharoen ◽  
Christine Böckmann
Keyword(s):  
Convergence Rate ◽  
Potential Problem ◽  
Regularization Parameter ◽  
Discrepancy Principle ◽  
Source Condition ◽  
Rate Analysis ◽  
Potential Problems ◽  
Ill Posed ◽  
Landweber Method ◽  
Convergence Rate Analysis

In this paper, we present the convergence rate analysis of the modified Landweber method under logarithmic source condition for nonlinear ill-posed problems. The regularization parameter is chosen according to the discrepancy principle. The reconstructions of the shape of an unknown domain for an inverse potential problem by using the modified Landweber method are exhibited.

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.

scholarly journals A LEPSKIJ-TYPE STOPPING-RULE FOR SIMPLIFIED ITERATIVELY REGULARIZED GAUSS-NEWTON METHOD

2017 ◽  
Author(s):  
Agah D. Garnadi
Keyword(s):  
Newton Method ◽  
Hilbert Spaces ◽  
Potential Problem ◽  
Initial Approximation ◽  
Stopping Rule ◽  
A Posteriori ◽  
Iterative Regularization ◽  
Ill Posed ◽  
Derivatives Of ◽  
Unknown Solution

Iterative regularization methods for nonlinear ill-posed equations of the form $ F(a)= y$, where $ F: D(F) \subset X \to Y$ is an operator between Hilbert spaces $ X $ and $ Y$, usually involve calculation of the Fr\'{e}chet derivatives of $ F$ at each iterate and at the unknown solution $ a^\sharp$. A modified form of the generalized Gauss-Newton method which requires the Fr\'{e}chet derivative of $F$ only at an initial approximation $ a_0$ of the solution $ a^\sharp$ as studied by Mahale and Nair \cite{MaNa:2k9}. This work studied an {\it a posteriori} stopping rule of Lepskij-type of the method. A numerical experiment from inverse source potential problem is demonstrated.

scholarly journals ON NUMERICAL REALIZATION OF QUASIOPTIMAL PARAMETER CHOICES IN (ITERATED) TIKHONOV AND LAVRENTIEV REGULARIZATION

2009 ◽  
Vol 14 (1) ◽  
pp. 99-108 ◽  
Author(s):  
Toomas Raus ◽  
Uno Hämarik
Keyword(s):  
Hilbert Spaces ◽  
Noise Level ◽  
Regularization Parameter ◽  
Numerical Realization ◽  
A Posteriori ◽  
Parameter Choice ◽  
Linear Combinations ◽  
Choice Alternative ◽  
Lavrentiev Regularization ◽  
Ill Posed

We consider linear ill‐posed problems in Hilbert spaces with noisy right hand side and given noise level. For approximation of the solution the Tikhonov method or the iterated variant of this method may be used. In self‐adjoint problems the Lavrentiev method or its iterated variant are used. For a posteriori choice of the regularization parameter often quasioptimal rules are used which require computing of additionally iterated approximations. In this paper we propose for parameter choice alternative numerical schemes, using instead of additional iterations linear combinations of approximations with different parameters.

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