Volume 38, pp. 233-257, 2011.

Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems

Stefan Kindermann


Minimization-based noise level-free parameter choice rules for the selection of the regularization parameter in linear ill-posed problems are studied. Abstract convergence results for spectral filter regularization operators using a qualitative condition on the (deterministic) data noise are proven. Furthermore, under source conditions on the exact solution, suboptimal convergence rates and, under certain additional regularity conditions, optimal order convergence rates are shown. The abstract results are examined in more detail for several known parameter choice rules: the quasi-optimality rules (both continuous and discrete) and the Hanke-Raus-rules, together with some specific regularization methods: Tikhonov regularization, Landweber iteration, and spectral cutoff.

Full Text (PDF) [324 KB], BibTeX

Key words

regularization, heuristic parameter choice rule, Hanke-Raus rule, quasi-optimality rule, L-curve method

AMS subject classifications

65J20, 47A52, 65J22

ETNA articles which cite this article

Vol. 39 (2012), pp. 437-463 Dirk A. Lorenz, Peter Maass, and Pham Q. Muoi: Gradient descent for Tikhonov functionals with sparsity constraints: Theory and numerical comparison of step size rules
Vol. 40 (2013), pp. 58-81 Stefan Kindermann: Discretization independent convergence rates for noise level-free parameter choice rules for the regularization of ill-conditioned problems
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
Vol. 58 (2023), pp. 348-377 Alessandro Buccini, Lucas Onisk, and Lothar Reichel: Range restricted iterative methods for linear discrete ill-posed problems
Vol. 60 (2024), pp. 327-350 Stefan Kindermann and Werner Zellinger: A short-term rational Krylov method for linear inverse problems
Vol. 61 (2024), pp. 66-91 Alessandro Buccini and Lothar Reichel: Software for limited memory restarted $l^p$-$l^q$ minimization methods using generalized Krylov subspaces

< Back