Journal of Inverse and Ill-posed Problems
Editor-in-Chief: Kabanikhin, Sergey I.
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On the choice of the regularization parameter in ill-posed problems with approximately given noise level of data
Citation Information: Journal of Inverse and Ill-posed Problems jiip. Volume 14, Issue 3, Pages 251–266, ISSN (Online) 1569-3953, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/156939406777340928,
- Published Online:
We consider regularization of linear ill-posed problems Au = ƒ in Hilbert spaces. Approximations u r to the solution u * can be constructed by the Tikhonov method or by the Lavrentiev method, by iterative or by other methods. We assume that instead of ƒ ∈ R(A) noisy data are available with the approximately given noise level δ: in process δ → 0 it holds || − ƒ||/δ ≤ c with unknown constant c. We propose a new a-posteriori rule for the choice of the regularization parameter r = r(δ) guaranteeing u r(δ) → u * for δ → 0. Note that such convergence is not guaranteed for the parameter choice given by the L-curve rule, by the GCV-rule, by the quasioptimality criterion and also for discrepancy principle ||Au r − || = bδ with b < c. The error estimates are given, which in case || − ƒ|| ≤ δ are quasioptimal and order-optimal.
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