The use of Morozov's discrepancy principle for Tikhonov regularization for solving nonlinear ill-posed problems

Computing ◽  
1993 ◽  
Vol 51 (1) ◽  
pp. 45-60 ◽  
Author(s):  
O. Scherzer
Keyword(s):  
Tikhonov Regularization ◽  
Discrepancy Principle ◽  
Ill Posed ◽  
Morozov’S Discrepancy Principle

scholarly journals Morozov’s Discrepancy Principle For The Tikhonov Regularization Of Exponentially Ill-Posed Problems

2008 ◽  
Vol 8 (1) ◽  
pp. 86-98 ◽  
Author(s):  
S.G. SOLODKY ◽  
A. MOSENTSOVA
Keyword(s):  
Tikhonov Regularization ◽  
Optimal Order ◽  
Discrepancy Principle ◽  
Operator Equations ◽  
Order Of Accuracy ◽  
Finite Dimensional ◽  
Dimensional Version ◽  
Ill Posed ◽  
Linear Operator Equations ◽  
Morozov’S Discrepancy Principle

Abstract The problem of approximate solution of severely ill-posed problems given in the form of linear operator equations of the first kind with approximately known right-hand sides was considered. We have studied a strategy for solving this type of problems, which consists in combinating of Morozov’s discrepancy principle and a finite-dimensional version of the Tikhonov regularization. It is shown that this combination provides an optimal order of accuracy on source sets

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 Optimisation in the regularisation ill-posed problems

1986 ◽  
Vol 28 (1) ◽  
pp. 114-133 ◽  
Author(s):  
A. R. Davies ◽  
R. S. Anderssen
Keyword(s):  
Maximum Likelihood ◽  
Tikhonov Regularization ◽  
Cross Validation ◽  
Discrepancy Principle ◽  
Asymptotic Estimates ◽  
Deconvolution Problem ◽  
Ill Posed ◽  
The Relationship ◽  
Asymptotic Analyses ◽  
Selection Of

We survey the role played by optimization in the choice of parameters for Tikhonov regularization of first-kind integral equations. Asymptotic analyses are presented for a selection of practical optimizing methods applied to a model deconvolution problem. These methods include the discrepancy principle, cross-validation and maximum likelihood. The relationship between optimality and regularity is emphasized. New bounds on the constants appearing in asymptotic estimates are presented.

scholarly journals DISCREPANCY SETS FOR COMBINED LEAST SQUARES PROJECTION AND TIKHONOV REGULARIZATION

2017 ◽  
Vol 22 (2) ◽  
pp. 202-212
Author(s):  
Teresa Reginska
Keyword(s):  
Least Squares ◽  
Tikhonov Regularization ◽  
Discrepancy Principle ◽  
Order Of Accuracy ◽  
Finite Dimensional ◽  
Regularization Parameters ◽  
Ill Posed ◽  
Data Error ◽  
Two Parameter ◽  
Regularized Solution

To solve a linear ill-posed problem, a combination of the finite dimensional least squares projection method and the Tikhonov regularization is considered. The dimension of the projection is treated as the second parameter of regularization. A two-parameter discrepancy principle defines a discrepancy set for any data error bound. The aim of the paper is to describe this set and to indicate its subset such that for regularization parameters from this subset the related regularized solution has the same order of accuracy as the Tikhonov regularization with the standard discrepancy principle but without any discretization.

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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