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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| Published in: | Applicable analysis Vol. 91; no. 5; pp. 1029 - 1044 |
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| Main Author: | |
| Format: | Journal Article |
| Language: | English |
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Abingdon
Taylor & Francis Group
01.05.2012
Taylor & Francis Ltd |
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| ISSN: | 0003-6811, 1563-504X |
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| Abstract | 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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| AbstractList | 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. 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. [PUBLICATION ABSTRACT] |
| Author | Flemming, Jens |
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| SubjectTerms | 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 |
| Title | Solution smoothness of ill-posed equations in Hilbert spaces: four concepts and their cross connections |
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