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U. Hmarik, T. Raus

On the choice of the regularization parameter in ill-posed problems with approximately given noise level of data

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.

Journal of Inverse and Ill-posed Problems, Walter de Gruyter

Print ISSN: 0928-0219
Volume: 14, 05/2006
Pages: 251 - 266

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