An iterative method for ill-posed problems with inequalities

1981 ◽  
Vol 25 (3) ◽  
pp. 101-105
Author(s):  
M. Z. Nashed
Keyword(s):  
Iterative Method ◽  
Ill Posed

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.

An iterative method to compute minimum norm solutions of ill-posed problems in Hilbert spaces

2019 ◽  
Vol 30 (5-6) ◽  
pp. 797-816
Author(s):  
Meisam Jozi ◽  
Saeed Karimi ◽  
Davod Khojasteh Salkuyeh
Keyword(s):  
Iterative Method ◽  
Hilbert Spaces ◽  
Minimum Norm ◽  
Ill Posed

A derivative-free iterative method for nonlinear ill-posed equations with monotone operators

2017 ◽  
Vol 25 (5) ◽  
pp. 543-551 ◽  
Author(s):  
Santhosh George ◽  
M. Thamban Nair
Keyword(s):  
Iterative Method ◽  
Convergence Analysis ◽  
Monotone Operator ◽  
Regularization Parameter ◽  
A Priori ◽  
Monotone Operators ◽  
Adaptive Procedure ◽  
Operator Equations ◽  
Derivative Free ◽  
Ill Posed

AbstractRecently, Semenova [12] considered a derivative free iterative method for nonlinear ill-posed operator equations with a monotone operator. In this paper, a modified form of Semenova’s method is considered providing simple convergence analysis under more realistic nonlinearity assumptions. The paper also provides a stopping rule for the iteration based on an a priori choice of the regularization parameter and also under the adaptive procedure considered by Pereverzev and Schock [11].

scholarly journals An iterative method for linear discrete ill-posed problems with box constraints

2007 ◽  
Vol 198 (2) ◽  
pp. 505-520 ◽  
Author(s):  
S. Morigi ◽  
L. Reichel ◽  
F. Sgallari ◽  
F. Zama
Keyword(s):  
Iterative Method ◽  
Box Constraints ◽  
Ill Posed

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.

scholarly journals Iterative Method to Solve a Data Completion Problem for Biharmonic Equation for Rectangular Domain

2017 ◽  
Vol 55 (1) ◽  
pp. 129-147
Author(s):  
Chakir Tajani ◽  
Houda Kajtih ◽  
Ali Daanoun
Keyword(s):  
Iterative Method ◽  
Biharmonic Equation ◽  
Rectangular Domain ◽  
Numerical Results ◽  
Cauchy Problems ◽  
Completion Problem ◽  
Data Completion ◽  
Ill Posed ◽  
Convergent Method ◽  
Direct Problems

AbstractIn this work, we are interested in a class of problems of great importance in many areas of industry and engineering. It is the invese problem for the biharmonic equation. It consists to complete the missing data on the inaccessible part from the measured data on the accessible part of the boundary. To solve this ill-posed problem, we opted for the alternative iterative method developed by Kozlov, Mazya and Fomin which is a convergent method for the elliptical Cauchy problems in general. The numerical implementation of the iterative algorithm is based on the application of the boundary element method (BEM) for a sequence of mixed well-posed direct problems. Numerical results are performed for a square domain showing the effectiveness of the algorithm by BEM to produce accurate and stable numerical results.

scholarly journals Kernel Optimization for Blind Motion Deblurring with Image Edge Prior

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Jing Wang ◽  
Ke Lu ◽  
Qian Wang ◽  
Jie Jia
Keyword(s):  
Iterative Method ◽  
Image Motion ◽  
Latent Image ◽  
Motion Deblurring ◽  
Image Prior ◽  
Ill Posed ◽  
Image Edge ◽  
Blur Kernel ◽  
Kernel Optimization ◽  
Blind Motion Deblurring

Image motion deblurring with unknown blur kernel is an ill-posed problem. This paper proposes a blind motion deblurring approach that solves blur kernel and the latent image robustly. For kernel optimization, an edge mask is used as an image prior to improve kernel update, then an edge selection mask is adopted to improve image update. In addition, an alternative iterative method is introduced to perform kernel optimization under a multiscale scheme. Moreover, for image restoration, a total-variation-(TV-) based algorithm is proposed to recover the latent image via nonblind deconvolution. Experimental results demonstrate that our method obtains accurate blur kernel and achieves better deblurring results than previous works.

A Quadratic Convergence Yielding Iterative Method for Nonlinear Ill-posed Operator Equations

2012 ◽  
Vol 12 (1) ◽  
pp. 32-45 ◽  
Author(s):  
Santhosh George ◽  
Atef Ibrahim Elmahdy
Keyword(s):  
Iterative Method ◽  
Operator Equation ◽  
Quadratic Convergence ◽  
Regularization Parameter ◽  
Optimal Order ◽  
Majorizing Sequence ◽  
Operator Equations ◽  
Order Error ◽  
Ill Posed ◽  
The Given

AbstractIn this paper, we consider an iterative method for the approximate solution of the nonlinear ill-posed operator equation Tx = y. The iteration procedure converges quadratically to the unique solution of the equation for the regularized approximation. It is known that (Tautanhahn (2002)) this solution converges to the solution of the given ill-posed operator equation. The convergence analysis and the stopping rule are based on a suitably constructed majorizing sequence. We show that the adaptive scheme considered by Perverzev and Schock (2005) for choosing the regularization parameter can be effectively used here for obtaining an optimal order error estimate.

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