scholarly journals Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems

2020 ◽  
Vol 0 (0) ◽  
pp. 1-28
Author(s):  
Ye Zhang ◽  
◽  
Bernd Hofmann ◽  
◽  
Keyword(s):  
Inverse Problems ◽  
Regularization Methods ◽  
Iterative Regularization ◽  
Ill Posed

Iterative Regularization Methods for Nonlinear Ill-Posed Problems

2008 ◽  
Author(s):  
Barbara Kaltenbacher ◽  
Andreas Neubauer ◽  
Otmar Scherzer
Keyword(s):  
Regularization Methods ◽  
Iterative Regularization ◽  
Ill Posed

scholarly journals On fractional asymptotical regularization of linear ill-posed problems in hilbert spaces

2019 ◽  
Vol 22 (3) ◽  
pp. 699-721 ◽  
Author(s):  
Ye Zhang ◽  
Bernd Hofmann
Keyword(s):  
Fractional Order ◽  
Hilbert Spaces ◽  
Regularization Method ◽  
Numerical Realization ◽  
Regularization Methods ◽  
Iterative Regularization ◽  
Operator Equations ◽  
Ill Posed ◽  
One Step ◽  
The One

Abstract In this paper, we study a fractional-order variant of the asymptotical regularization method, called Fractional Asymptotical Regularization (FAR), for solving linear ill-posed operator equations in a Hilbert space setting. We assign the method to the general linear regularization schema and prove that under certain smoothness assumptions, FAR with fractional order in the range (1, 2) yields an acceleration with respect to comparable order optimal regularization methods. Based on the one-step Adams-Moulton method, a novel iterative regularization scheme is developed for the numerical realization of FAR. Two numerical examples are given to show the accuracy and the acceleration effect of FAR.

scholarly journals Modern regularization methods for inverse problems

Acta Numerica ◽  
2018 ◽  
Vol 27 ◽  
pp. 1-111 ◽  
Author(s):  
Martin Benning ◽  
Martin Burger
Keyword(s):  
Inverse Problems ◽  
Future Research ◽  
Regularization Methods ◽  
Comprehensive Overview ◽  
Linear Inverse Problems ◽  
Recent Interest ◽  
Methods And Techniques ◽  
Ill Posed ◽  
Nonlinear Regularization ◽  
Statistical Inverse Problems

Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and allow a robust approximation of ill-posed (pseudo-) inverses. In the last two decades interest has shifted from linear to nonlinear regularization methods, even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this shift towards modern nonlinear regularization methods, including their analysis, applications and issues for future research.In particular we will discuss variational methods and techniques derived from them, since they have attracted much recent interest and link to other fields, such as image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions and learning theory.

Accuracy estimates of Gauss–Newton-type iterative regularization methods for nonlinear equations with operators having normally solvable derivative at the solution

2016 ◽  
Vol 24 (4) ◽  
Author(s):  
Mikhail Y. Kokurin
Keyword(s):  
Hilbert Spaces ◽  
Total Error ◽  
A Priori ◽  
Stopping Rules ◽  
Regularization Methods ◽  
Iterative Regularization ◽  
Operator Equations ◽  
Ill Posed ◽  
Irregular Operator ◽  
The Right

AbstractWe investigate a class of iterative regularization methods for solving nonlinear irregular operator equations in Hilbert spaces. The operator of an equation is supposed to have a normally solvable derivative at the desired solution. The operators and right parts of equations can be given with errors. A priori and a posteriori stopping rules for the iterations are analyzed. We prove that the accuracy of delivered approximations is proportional to the total error level in the operator and the right part of an equation. The obtained results improve known accuracy estimates for the class of iterative regularization methods, as applied to general irregular operator equations. The results also extend previous similar estimates related to regularization methods for linear ill-posed equations with normally solvable operators.

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