Convergence rates for Morozov's discrepancy principle using variational inequalities

Abstract

We derive convergence rates for Tikhonov-type regularization with convex penalty terms, where the regularization parameter is chosen according to Morozov's discrepancy principle and variational inequalities are used to generalize classical source and nonlinearity conditions. Rates are obtained first with respect to the Bregman distance and a Taylor-type distance and those results are combined to derive rates in norm and the penalty term topology.For the special case of the sparsity promoting weighted ℓp-norms as penalty terms and for a searched-for solution, which is known to be sparse, the above results give convergence rates of up to linear order.

Wavelet methods for a weighted sparsity penalty for region of interest tomography

“…We may add that results for different parameter choice rules do exist, e.g., for the discrepancy principle [1]. We also would like to mention convergence speed results, which require, e.g., source conditions for the solution [2,3,8,14,26,27,41,51]. Convergence and convergence rates for sparse regularization in a Bayes setup have been recently investigated in [25].…”

Convergence rates inℓ1-regularization when the basis is not smooth enough

“…For successful application of 1 -regularization, existence of minimizers x δ α to (1) and their stability with respect to perturbations in the data y δ have to be ensured. Further, by choosing the regularization parameter α > 0 in dependence on the noise level δ and the given data y δ , one has to guarantee that corresponding minimizers converge to a solution x † of (2) if the noise level goes to zero.…”

Regularization Methods for Ill-Posed Problems

“…However, under the following assumption, the existence of a solution x 2 D.F / to (1) implies the existence of an x -minimum norm solution (see [107,Lemma 3.2] and if (1) has a solution in D.F /; ı n ! 0 is a sequence of noise levels with corresponding data y n D y ı n such that ky n yk Ä ı n , then every associated sequence fx n g of minimizers to (6) has a convergent subsequence, and all limit elements are x -minimum norm solutions x of (1).…”

“…Please refer to the papers [6,32,44,51,53,63,99] for a further discussion of alternative misfit functionals S. In most cases convex penalty functionals R are preferred. An important class of penalty functionals form the norm powers R.x/ WD kxk q Q X ; q > 0; x 2 Q X; where as an alternative to the standard case X D Q X the space Q X can also be chosen as a dense subspace of X with stronger norm, e.g., X D L q .…”