scholarly journals On tensor GMRES and Golub-Kahan methods via the T-product for color image processing

2021 ◽  
Vol 37 ◽  
pp. 524-543
Author(s):  
Mohamed El Guide ◽  
Alaa El Ichi ◽  
Khalide Jbilou ◽  
Rachid Sadaka
Keyword(s):  
Tikhonov Regularization ◽  
Cross Validation ◽  
Krylov Subspace ◽  
Color Image ◽  
Regularization Parameter ◽  
Color Image Processing ◽  
Regularization Technique ◽  
Image Sequence Processing ◽  
Ill Posed ◽  
Selection Of

The present paper is concerned with developing tensor iterative Krylov subspace methods to solve large multi-linear tensor equations. We use the T-product for two tensors to define tensor tubal global Arnoldi and tensor tubal global Golub-Kahan bidiagonalization algorithms. Furthermore, we illustrate how tensor-based global approaches can be exploited to solve ill-posed problems arising from recovering blurry multichannel (color) images and videos, using the so-called Tikhonov regularization technique, to provide computable approximate regularized solutions. We also review a generalized cross-validation and discrepancy principle type of criterion for the selection of the regularization parameter in the Tikhonov regularization. Applications to image sequence processing are given to demonstrate the efficiency of the algorithms.

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 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.

scholarly journals COMPLEXITY ESTIMATES FOR SEVERELY ILL-POSED PROBLEMS UNDER A POSTERIORI SELECTION OF REGULARIZATION PARAMETER

2017 ◽  
Vol 22 (3) ◽  
pp. 283-299
Author(s):  
Sergii G. Solodky ◽  
Ganna L. Myleiko ◽  
Evgeniya V. Semenova
Keyword(s):  
Integral Equations ◽  
Regularization Parameter ◽  
Efficient Algorithms ◽  
Optimal Order ◽  
A Posteriori ◽  
Order Of Accuracy ◽  
Tikhonov Method ◽  
Discrete Information ◽  
Ill Posed ◽  
Selection Of

In the article the authors developed two efficient algorithms for solving severely ill-posed problems such as Fredholm’s integral equations. The standard Tikhonov method is applied as a regularization. To select a regularization parameter we employ two different a posteriori rules, namely, discrepancy and balancing principles. It is established that proposed strategies not only achieved optimal order of accuracy on the class of problems under consideration, but also they are economical in the sense of used discrete information.

scholarly journals Global Quasi-Minimal Residual Method for Image Restoration

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Jun Liu ◽  
Ting-Zhu Huang ◽  
Xiao-Guang Lv ◽  
Hao Xu ◽  
Xi-Le Zhao
Keyword(s):  
Iterative Method ◽  
Image Restoration ◽  
Tikhonov Regularization ◽  
Kronecker Product ◽  
Gmres Method ◽  
Regularization Technique ◽  
Minimal Residual Method ◽  
Ill Posed ◽  
Qmr Method ◽  
Minimal Residual

The global quasi-minimal residual (QMR) method is a popular iterative method for the solution of linear systems with multiple right-hand sides. In this paper, we consider the application of the global QMR method to classical ill-posed problems arising from image restoration. Since the scale of the problem is usually very large, the computations with the blurring matrix can be very expensive. In this regard, we use a Kronecker product approximation of the blurring matrix to benefit the computation. In order to reduce the disturbance of noise to the solution, the Tikhonov regularization technique is adopted to produce better approximation of the desired solution. Numerical results show that the global QMR method outperforms the classic CGLS method and the global GMRES method.

Lavrentiev Regularization and Balancing Principle for Solving Ill-Posed Problems with Monotone Operators

2010 ◽  
Vol 10 (4) ◽  
pp. 444-454 ◽  
Author(s):  
E.V. Semenova
Keyword(s):  
Regularization Parameter ◽  
Monotone Operators ◽  
Smooth Solutions ◽  
Lavrentiev Regularization ◽  
Ill Posed ◽  
Balancing Principle ◽  
Selection Of

AbstractThe paper considers a method for solving nonlinear ill-posed problems with monotone operators. The approach combines the Lavrentiev method, the fixedpoint method, and the balancing principle for selection of the regularization parameter. The method’s optimality has been proved for some set of smooth solutions. A test example proves the efficiency of the proposed method.

A fast multilevel iteration method for solving linear ill-posed integral equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hongqi Yang ◽  
Rong Zhang
Keyword(s):  
Approximate Solution ◽  
Linear System ◽  
Integral Equations ◽  
Tikhonov Regularization ◽  
Iteration Method ◽  
Noise Level ◽  
Regularization Parameter ◽  
Ill Posed ◽  
Linear Integral ◽  
Linear Integral Equations

Abstract We propose a new concept of noise level: R ⁢ ( K * ) \mathcal{R}(K^{*}) -noise level for ill-posed linear integral equations in Tikhonov regularization, which extends the range of regularization parameter. This noise level allows us to choose a more suitable regularization parameter. Moreover, we also analyze error estimates of the approximate solution with respect to this noise level. For ill-posed integral equations, finding fast and effective numerical methods is a challenging problem. For this, we formulate a matrix truncated strategy based on multiscale Galerkin method to generate the linear system of Tikhonov regularization for ill-posed linear integral equations, which greatly reduce the computational complexity. To further reduce the computational cost, a fast multilevel iteration method for solving the linear system is established. At the same time, we also prove convergence rates of the approximate solution obtained by this fast method with respect to the R ⁢ ( K * ) \mathcal{R}(K^{*}) -noise level under the balance principle. By numerical results, we show that R ⁢ ( K * ) \mathcal{R}(K^{*}) -noise level is very useful and the proposed method is a fast and effective method, respectively.

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