Tikhonov regularization of nonlinear ill-posed equations under general source condition

2007 ◽  
Vol 15 (8) ◽  
Author(s):  
Pallavi Mahale ◽  
M. Thamban Nair
Keyword(s):  
Tikhonov Regularization ◽  
Source Condition ◽  
Ill Posed ◽  
General Source

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.

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.

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 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.

Simplified Iterated Lavrentiev Regularization for Nonlinear Ill-Posed Monotone Operator Equations

2017 ◽  
Vol 17 (2) ◽  
pp. 269-285 ◽  
Author(s):  
Pallavi Mahale
Keyword(s):  
Exact Solution ◽  
Monotone Operator ◽  
Initial Approximation ◽  
Fréchet Derivative ◽  
Optimal Error Estimate ◽  
Source Condition ◽  
Frechet Derivative ◽  
Lavrentiev Regularization ◽  
Ill Posed ◽  
General Source

AbstractMahale and Nair [12] considered an iterated form of Lavrentiev regularization for obtaining stable approximate solutions for ill-posed nonlinear equations of the form ${F(x)=y}$, where ${F:D(F)\subseteq X\to X}$ is a nonlinear monotone operator and X is a Hilbert space. They considered an a posteriori strategy to find a stopping index which not only led to the convergence of the method, but also gave an order optimal error estimate under a general source condition. However, the iterations defined in [12] require calculation of Fréchet derivatives at each iteration. In this paper, we consider a simplified version of the iterated Lavrentiev regularization which will involve calculation of the Fréchet derivative only at the point ${x_{0}}$, i.e., at the initial approximation of the exact solution ${x^{\dagger}}$. Moreover, the general source condition and stopping rule which we use in this paper involve calculation of the Fréchet derivative at the point ${x_{0}}$, instead at the unknown exact solution ${x^{\dagger}}$ as in [12].

scholarly journals ON MOROZOV’S DISCREPANCY PRINCIPLE FOR NONLINEAR ILL-POSED EQUATIONS

2009 ◽  
Vol 79 (2) ◽  
pp. 337-342 ◽  
Author(s):  
M. T. NAIR
Keyword(s):  
Tikhonov Regularization ◽  
Nonlinear Problems ◽  
Discrepancy Principle ◽  
Parameter Choice ◽  
Operator Equations ◽  
Ill Posed ◽  
Morozov’S Discrepancy Principle ◽  
General Source

AbstractMorozov’s discrepancy principle is one of the simplest and most widely used parameter choice strategies in the context of regularization of ill-posed operator equations. Although many authors have considered this principle under general source conditions for linear ill-posed problems, such study for nonlinear problems is restricted to only a few papers. The aim of this paper is to apply Morozov’s discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems under general source conditions.

scholarly journals A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1685-1697
Author(s):  
Zhenyu Zhao ◽  
Lei You ◽  
Zehong Meng
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
Tikhonov Regularization Method ◽  
Hermite Expansion ◽  
The Cauchy Problem ◽  
Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

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