Solution smoothness of ill-posed equations in Hilbert spaces: four concepts and their cross connections

Numerical solution of ill-posed operator equations requires regularization techniques. The convergence of regularized solutions to the exact solution can be usually guaranteed, but to also obtain estimates for the speed of convergence one has to exploit some kind of smoothness of the exact solution....

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Bibliographic Details
Published in:Applicable analysis Vol. 91; no. 5; pp. 1029 - 1044
Main Author: Flemming, Jens
Format: Journal Article
Language:English
Published: Abingdon Taylor & Francis Group 01.05.2012
Taylor & Francis Ltd
Subjects:
Approximation
Convergence
Euclidean space
Exact solutions
Hilbert space
ill-posed problem
Inequalities
Mathematical analysis
Mathematical functions
Mathematical models
Mathematical problems
Numerical analysis
Smoothness
source condition
variational inequality
ISSN:0003-6811, 1563-504X
Online Access:Get full text
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Summary:Numerical solution of ill-posed operator equations requires regularization techniques. The convergence of regularized solutions to the exact solution can be usually guaranteed, but to also obtain estimates for the speed of convergence one has to exploit some kind of smoothness of the exact solution. We consider four such smoothness concepts in a Hilbert space setting: source conditions, approximate source conditions, variational inequalities, and approximate variational inequalities. Besides some new auxiliary results on variational inequalities the equivalence of the last three concepts is shown. In addition, it turns out that the classical concept of source conditions and the modern concept of variational inequalities are connected via Fenchel duality.
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ISSN:0003-6811
1563-504X
DOI:10.1080/00036811.2011.563736