scholarly journals Implementations of range restricted iterative methods for linear discrete ill-posed problems

2012 ◽  
Vol 436 (10) ◽  
pp. 3974-3990 ◽  
Author(s):  
A. Neuman ◽  
L. Reichel ◽  
H. Sadok
Keyword(s):  
Iterative Methods ◽  
Ill Posed

M. A. Krasnosel'skii Theorem and Iterative Methods for Solving Ill-Posed Linear Problems with a Self-Adjoint Operator

2015 ◽  
Vol 15 (3) ◽  
pp. 373-389
Author(s):  
Oleg Matysik ◽  
Petr Zabreiko
Keyword(s):  
Hilbert Space ◽  
Iterative Methods ◽  
Critical Case ◽  
Adjoint Operator ◽  
Successive Approximations ◽  
Operator Equations ◽  
Ill Posed ◽  
Linear Problems ◽  
Adjoint Operators ◽  
Linear Operator Equations

AbstractThe paper deals with iterative methods for solving linear operator equations ${x = Bx + f}$ and ${Ax = f}$ with self-adjoint operators in Hilbert space X in the critical case when ${\rho (B) = 1}$ and ${0 \in \operatorname{Sp} A}$. The results obtained are based on a theorem by M. A. Krasnosel'skii on the convergence of the successive approximations, their modifications and refinements.

scholarly journals Optimization Learning: Perspective, Method, and Applications

2020 ◽  
Author(s):  
Risheng Liu
Keyword(s):  
Inverse Problems ◽  
Theoretical Analysis ◽  
Iterative Methods ◽  
State Of The Art ◽  
Learning Paradigm ◽  
Rigorous Analysis ◽  
The Core ◽  
Globally Convergent ◽  
Ill Posed ◽  
Art Performance

Numerous tasks at the core of statistics, learning, and vision areas are specific cases of ill-posed inverse problems. Recently, learning-based (e.g., deep) iterative methods have been empirically shown to be useful for these problems. Nevertheless, integrating learnable structures into iterations is still a laborious process, which can only be guided by intuitions or empirical insights. Moreover, there is a lack of rigorous analysis of the convergence behaviors of these reimplemented iterations, and thus the significance of such methods is a little bit vague. We move beyond these limits and propose a theoretically guaranteed optimization learning paradigm, a generic and provable paradigm for nonconvex inverse problems, and develop a series of convergent deep models. Our theoretical analysis reveals that the proposed optimization learning paradigm allows us to generate globally convergent trajectories for learning-based iterative methods. Thanks to the superiority of our framework, we achieve state-of-the-art performance on different real applications.

scholarly journals On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations

2010 ◽  
Vol 4 (3) ◽  
pp. 335-350 ◽  
Author(s):  
Johann Baumeister ◽  
◽  
Barbara Kaltenbacher ◽  
Antonio Leitão ◽  
◽  
...  
Keyword(s):  
Iterative Methods ◽  
Levenberg Marquardt ◽  
Ill Posed

Iterative Methods for Ill-Posed Problems

2011 ◽  
Author(s):  
Anatoly B. Bakushinsky ◽  
Mihail Yu. Kokurin ◽  
Alexandra Smirnova
Keyword(s):  
Iterative Methods ◽  
Ill Posed

Automatic stopping rule for iterative methods in discrete ill-posed problems

2014 ◽  
Vol 34 (3) ◽  
pp. 1175-1197 ◽  
Author(s):  
Leonardo S. Borges ◽  
Fermín S. Viloche Bazán ◽  
Maria C. C. Cunha
Keyword(s):  
Iterative Methods ◽  
Stopping Rule ◽  
Ill Posed
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