scholarly journals Lavrentiev Regularization for Linear Ill-Posed Problems under General Source Conditions

2004 ◽  
pp. 167-185 ◽  
Author(s):  
M. Thamban Nair ◽  
U. Tautenhahn
Keyword(s):  
Lavrentiev Regularization ◽  
Ill Posed ◽  
General Source

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 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 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 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 A Preconditioned Richardson Regularization for the Data Completion Problem and the Kozlov-Maz’ya-Fomin Method

2010 ◽  
Vol Volume 13 - 2010 - Special... ◽  
Author(s):  
Duc Thang Du ◽  
Faten Jelassi
Keyword(s):  
Inverse Problems ◽  
Stopping Rules ◽  
Completion Problem ◽  
Variational Framework ◽  
Data Completion ◽  
International Audience ◽  
Ill Posed ◽  
General Source ◽  
Richardson Iterative Method ◽  
Math Phys

International audience Using a preconditioned Richardson iterative method as a regularization to the data completion problem is the aim of the contribution. The problem is known to be exponentially ill posed that makes its numerical treatment a hard task. The approach we present relies on the Steklov-Poincaré variational framework introduced in [Inverse Problems, vol. 21, 2005]. The resulting algorithm turns out to be equivalent to the Kozlov-Maz’ya-Fomin method in [Comp. Math. Phys., vol. 31, 1991]. We conduct a comprehensive analysis on the suitable stopping rules that provides some optimal estimates under the General Source Condition on the exact solution. Some numerical examples are finally discussed to highlight the performances of the method. L’objectif est d’utiliser une méthode itérative de Richardson préconditionnée comme une technique de régularisation pour le problème de complétion de données. Le problème est connu pour être sévèrement mal posé qui rend son traitement numérique ardu. L’approche adoptée est basée sur le cadre variationnel de Steklov-Poincaré introduit dans [Inverse Problems, vol. 21, 2005].L’algorithme obtenu s’avère être équivalent à celui de Kozlov-Maz’ya-Fomin parû dans [Comp. Math. Phys., vol. 31, 1991]. Nous menons une analyse complète pour le choix du critère d’arrêt, et établissons des estimations optimales sous les Conditions Générale de Source sur la solution exacte. Nous discutons, enfin, quelques exemples numériques qui confortent les pertinence de la méthode.

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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