Convergence rates in constrained Tikhonov regularization: equivalence of projected source conditions and variational inequalities
In this paper, we enlighten the role of variational inequalities for obtaining convergence rates in Tikhonov regularization of nonlinear ill-posed problems with convex penalty functionals under convexity constraints in Banach spaces. Variational inequalities are able to cover solution smoothness and...
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| Veröffentlicht in: | Inverse problems 2011-08, Vol.27 (8), p.085001-11 |
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| container_title | Inverse problems |
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| creator | Flemming, Jens Hofmann, Bernd |
| description | In this paper, we enlighten the role of variational inequalities for obtaining convergence rates in Tikhonov regularization of nonlinear ill-posed problems with convex penalty functionals under convexity constraints in Banach spaces. Variational inequalities are able to cover solution smoothness and the structure of nonlinearity in a uniform manner, not only for unconstrained but, as we indicate, also for constrained Tikhonov regularization. In this context, we extend the concept of projected source conditions already known in Hilbert spaces to Banach spaces, and we show in the main theorem that such projected source conditions are to some extent equivalent to certain variational inequalities. The derived variational inequalities immediately yield convergence rates measured by Bregman distances. |
| doi_str_mv | 10.1088/0266-5611/27/8/085001 |
| format | Article |
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| source | IOP Publishing Journals; Institute of Physics (IOP) Journals - HEAL-Link |
| subjects | Banach space Constraints Convergence Equivalence Exact sciences and technology Inequalities Inverse problems Nonlinearity Physics Regularization Smoothness |
| title | Convergence rates in constrained Tikhonov regularization: equivalence of projected source conditions and variational inequalities |
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