On the Parameter Choice in the Multilevel Augmentation Method

2020 ◽  
Vol 20 (3) ◽  
pp. 555-571
Author(s):  
Suhua Yang ◽  
Xingjun Luo ◽  
Chunmei Zeng ◽  
Zhihai Xu ◽  
Wenyu Hu
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Method ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
Tikhonov Regularization Method ◽  
Fredholm Integral Equations ◽  
Parameter Choice ◽  
Choice Strategy ◽  
Ill Posed ◽  
Parameter Choice Strategy

AbstractIn this paper, we apply the multilevel augmentation method for solving ill-posed Fredholm integral equations of the first kind via iterated Tikhonov regularization method. The method leads to fast solutions of the discrete regularization methods for the equations. The convergence rates of iterated Tikhonov regularization are achieved by using a modified parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method.

A parameter choice strategy for a multilevel augmentation method in iterated Lavrentiev regularization

2018 ◽  
Vol 26 (2) ◽  
pp. 153-170 ◽  
Author(s):  
Chunmei Zeng ◽  
Xingjun Luo ◽  
Suhua Yang ◽  
Fanchun Li
Keyword(s):  
Integral Equation ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
Parameter Choice ◽  
Lavrentiev Regularization ◽  
Choice Strategy ◽  
Ill Posed ◽  
Parameter Choice Strategy

AbstractIn this paper we apply the multilevel augmentation method to solve an ill-posed integral equation via the iterated Lavrentiev regularization. This method leads to fast solutions of discrete iterated Lavrentiev regularization. The convergence rates of the iterated Lavrentiev regularization are achieved by using a certain parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method.

Multiscale Galerkin methods for the nonstationary iterated Tikhonov method with a modified posteriori parameter selection

2018 ◽  
Vol 26 (1) ◽  
pp. 109-120
Author(s):  
Xingjun Luo ◽  
Zhaofu Ouyang ◽  
Chunmei Zeng ◽  
Fanchun Li
Keyword(s):  
Convergence Rates ◽  
Regularization Parameter ◽  
Fredholm Integral Equations ◽  
Galerkin Methods ◽  
A Posteriori ◽  
Parameter Choice ◽  
Compression Technique ◽  
Choice Strategy ◽  
Optimal Convergence Rates ◽  
Parameter Choice Strategy

AbstractIn this paper, we consider a fast multiscale Galerkin method with compression technique for solving Fredholm integral equations of the first kind via the nonstationary iterated Tikhonov regularization. A modified a posteriori regularization parameter choice strategy is established, which leads to optimal convergence rates.

THE MODIFIED TIKHONOV REGULARIZATION METHOD WITH THE SMOOTHED TOTAL VARIATION FOR SOLVING THE LINEAR ILL-POSED PROBLEMS

2021 ◽  
Vol 9 (2) ◽  
pp. 88-100
Author(s):  
Vladimir Vasin ◽  
◽  
Vladimir Belyaev
Keyword(s):  
Linear Operator ◽  
Total Variation ◽  
Operator Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Numerical Experiments ◽  
Approximate Solutions ◽  
Tikhonov Regularization Method ◽  
Smooth Component ◽  
Ill Posed

We investigate a linear operator equation of the first kind that is ill-posed in the Hadamard sence. It is assumed that its solution is representable as a sum of smooth and discontinuous components. To construct a stable approximate solutions, we use the modified Tikhonov method with the stabilizing functional as a sum of the Lebesgue norm for the smooth component and a smoothed BV-norm for the discontinuous component. Theorems of exis- tence, uniqueness, and convergence both the regularized solutions and its finite-dimentional approximations are proved. Also, results of numerical experiments are presented.

scholarly journals Augmented Arnoldi-Tikhonov Regularization Methods for Solving Large-Scale Linear Ill-Posed Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Yiqin Lin ◽  
Liang Bao ◽  
Yanhua Cao
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Method ◽  
Numerical Experiments ◽  
Large Scale ◽  
Krylov Subspace ◽  
Dimensional Subspace ◽  
Tikhonov Regularization Method ◽  
Rough Approximation ◽  
Ill Posed ◽  
Low Dimensional

We propose an augmented Arnoldi-Tikhonov regularization method for the solution of large-scale linear ill-posed systems. This method augments the Krylov subspace by a user-supplied low-dimensional subspace, which contains a rough approximation of the desired solution. The augmentation is implemented by a modified Arnoldi process. Some useful results are also presented. Numerical experiments illustrate that the augmented method outperforms the corresponding method without augmentation on some real-world examples.

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.

The Regularization Method Based on Complex Collinearity Diagnostics and Metrics

2013 ◽  
Vol 416-417 ◽  
pp. 1393-1398
Author(s):  
Chao Zhong Ma ◽  
Yong Wei Gu ◽  
Ji Fu ◽  
Yuan Lu Du ◽  
Qing Ming Gui
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Measurement Data ◽  
Tikhonov Regularization Method ◽  
Parameter Determination ◽  
Selection Methods ◽  
Ill Posed ◽  
Collinearity Diagnostics ◽  
Regularization Matrix

In a large number of measurement data processing, the ill-posed problem is widespread. For such problems, this paper introduces the solution of ill-posed problem of the unity of expression and Tikhonov regularization method, and then to re-collinearity diagnostics and metrics based on proposed based on complex collinearity diagnostics and the metric regularization method is given regularization matrix selection methods and regularization parameter determination formulas. Finally, it uses a simulation example to verify the effectiveness of the method.

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