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.

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.

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].

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 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.

On The Analog Of The Monotone Error Rule For Parameter Choice In The (Iterated) Lavrentiev Regularization

2008 ◽  
Vol 8 (3) ◽  
pp. 237-252 ◽  
Author(s):  
U HAMARIK ◽  
R. PALM ◽  
T. RAUS
Keyword(s):  
Hilbert Spaces ◽  
Noise Level ◽  
Regularization Parameter ◽  
Optimal Error Estimates ◽  
Discrepancy Principle ◽  
Parameter Choice ◽  
Optimal Error ◽  
Numerical Examples ◽  
Lavrentiev Regularization ◽  
Ill Posed

AbstractWe consider linear ill-posed problems in Hilbert spaces with a noisy right hand side and a given noise level. To solve non-self-adjoint problems by the (it-erated) Tikhonov method, one effective rule for choosing the regularization parameter is the monotone error rule (Tautenhahn and Hamarik, Inverse Problems, 1999, 15, 1487– 1505). In this paper we consider the solution of self-adjoint problems by the (iterated) Lavrentiev method and propose for parameter choice an analog of the monotone error rule. We prove under certain mild assumptions the quasi-optimality of the proposed rule guaranteeing convergence and order optimal error estimates. Numerical examples show for the proposed rule and its modifications much better performance than for the modified discrepancy principle.

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 Heating source localization in a reduced time

2016 ◽  
Vol 26 (3) ◽  
pp. 623-640 ◽  
Author(s):  
Sara Beddiaf ◽  
Laurent Autrique ◽  
Laetitia Perez ◽  
Jean-Claude Jolly
Keyword(s):  
Heat Conduction ◽  
Source Localization ◽  
Conjugate Gradient ◽  
Regularization Method ◽  
Three Dimensional ◽  
Gradient Algorithm ◽  
Iterative Regularization ◽  
Heating Source ◽  
Ill Posed ◽  
Reduced Time

Abstract Inverse three-dimensional heat conduction problems devoted to heating source localization are ill posed. Identification can be performed using an iterative regularization method based on the conjugate gradient algorithm. Such a method is usually implemented off-line, taking into account observations (temperature measurements, for example). However, in a practical context, if the source has to be located as fast as possible (e.g., for diagnosis), the observation horizon has to be reduced. To this end, several configurations are detailed and effects of noisy observations are investigated.

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.

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