Some convergence results of a modified asymptotical regularization gradient method for nonlinear ill-posed operator equation

2010 ◽  
Vol 87 (12) ◽  
pp. 2811-2822
Author(s):  
Yehui Peng ◽  
Heying Feng ◽  
Aijun Bu
Keyword(s):  
Operator Equation ◽  
Gradient Method ◽  
Convergence Results ◽  
Ill Posed

scholarly journals Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems

2021 ◽  
Author(s):  
Radu Boţ ◽  
Guozhi Dong ◽  
Peter Elbau ◽  
Otmar Scherzer
Keyword(s):  
Linear Operator ◽  
Operator Equation ◽  
Convex Analysis ◽  
Convergence Rates ◽  
Linear Operator Equation ◽  
Optimal Convergence Rates ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Given ◽  
Theoretical Results

AbstractRecently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

scholarly journals Convergence of the gradient method for ill-posed problems

2017 ◽  
Vol 11 (4) ◽  
pp. 703-720 ◽  
Author(s):  
Stefan Kindermann ◽  
Keyword(s):  
Gradient Method ◽  
Ill Posed

Theory Study for Moving Force Identification

2012 ◽  
Vol 476-478 ◽  
pp. 2292-2295
Author(s):  
Zhen Chen
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Noise Immunity ◽  
Identification Accuracy ◽  
Force Identification ◽  
Moving Force ◽  
Study Results ◽  
Ill Posed ◽  
Pre Treatment

Three identification methods, the time domain method (TDM)、the conjugate gradient method (CGM)and the pre-treatment conjugate gradient method (PCGM) are employed for moving force identification. Related research shows that the PCGM have higher identification accuracy and robust noise immunity as well as producing an acceptable solution to ill-posed cases to some extent when they are used to identify the moving force. However, the pre-treatment matrix is very important to the PCGM because it affects the identification accuracy and robust noise immunity as well as ill-posed cases to some extent. The theory study results are practical significant to selection properly pre-treatment matrix.

Error minimizing relaxation strategies in Landweber and Kaczmarz type iterations

2017 ◽  
Vol 25 (1) ◽  
Author(s):  
Touraj Nikazad ◽  
Mokhtar Abbasi ◽  
Tommy Elfving
Keyword(s):  
Image Reconstruction ◽  
Linear Systems ◽  
Operator Theory ◽  
Convex Constraints ◽  
Advantages And Disadvantages ◽  
Exact Data ◽  
Convergence Results ◽  
Ill Posed ◽  
Image Reconstruction From Projections

AbstractWe study error minimizing relaxation (EMR) strategies for use in Landweber and Kaczmarz type iterations applied to linear systems with or without convex constraints. Convergence results based on operator theory are given, assuming exact data. The advantages and disadvantages of these relaxation strategies on a noisy and ill-posed problem are illustrated using examples taken from the field of image reconstruction from projections. We also consider combining EMR with penalization.

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