Volume 40, pp. 58-81, 2013.

Discretization independent convergence rates for noise level-free parameter choice rules for the regularization of ill-conditioned problems

Stefan Kindermann


We develop a convergence theory for noise level-free parameter choice rules for Tikhonov regularization of finite-dimensional, linear, ill-conditioned problems. In particular, we derive convergence rates with bounds that do not depend on troublesome parameters such as the small singular values of the system matrix. The convergence analysis is based on specific qualitative assumptions on the noise, the noise conditions, and on certain regularity conditions. Furthermore, we derive several sufficient noise conditions both in the discrete and infinite-dimensional cases. This leads to important conclusions for the actual implementation of such rules in practice. For instance, we show that for the case of random noise, the regularization parameter can be found by minimizing a parameter choice functional over a subinterval of the spectrum (whose size depends on the smoothing properties of the forward operator), yielding discretization independent convergence rate estimates, which are of optimal order under regularity assumptions for the exact solution.

Full Text (PDF) [286 KB], BibTeX

Key words

regularization, parameter choice rule, Hanke-Raus rule, quasioptimality rule, generalized cross validation

AMS subject classifications

65J20, 47A52, 65J22

Links to the cited ETNA articles

[10]Vol. 38 (2011), pp. 233-257 Stefan Kindermann: Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems

ETNA articles which cite this article

Vol. 43 (2014-2015), pp. 223-243 Lothar Reichel and Xuebo Yu: Matrix decompositions for Tikhonov regularization
Vol. 48 (2018), pp. 435-449 Jenny Niebsch and Ronny Ramlau: Frequency-dependent reconstruction of imbalances
Vol. 53 (2020), pp. 217-238 Stefan Kindermann and Kemal Raik: A simplified L-curve method as error estimator
Vol. 57 (2022), pp. 216-241 Simon Hubmer, Ekaterina Sherina, Stefan Kindermann, and Kemal Raik: A numerical comparison of some heuristic stopping rules for nonlinear Landweber iteration

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