No abstract

Variational sparsity regularization based on 1 -norms and other nonlinear functionals has gained enormous attention recently, both with respect to its applications and its mathematical analysis. A focus in regularization theory has been to develop error estimation in terms of regularization parameter and noise strength. For this sake, specific error measures such as Bregman distances and specific conditions on the solution such as source conditions or variational inequalities have been developed and used. In this paper we provide, for a certain class of ill-posed linear operator equations, a convergence analysis that works for solutions that are not completely sparse, but have a fast-decaying nonzero part. This case is not covered by standard source conditions, but surprisingly can be treated with an appropriate variational inequality. As a consequence, the paper also provides the first examples where the variational inequality approach, which was often believed to be equivalent to appropriate source conditions, can indeed go farther than the latter.

We show that the convergence rate of ℓ 1 -regularization for linear ill-posed equations is always O(δ) if the exact solution is sparse and if the considered operator is injective and weak*-to-weak continuous. Under the same assumptions convergence rates in case of non-sparse solutions are proven. The results base on the fact that certain sourcetype conditions used in the literature for proving convergence rates are automatically satisfied.

In this paper, we enlighten the role of variational inequalities for obtaining convergence rates in Tikhonov regularization of nonlinear ill-posed problems with convex penalty functionals under convexity constraints in Banach spaces. Variational inequalities are able to cover solution smoothness and the structure of nonlinearity in a uniform manner, not only for unconstrained but, as we indicate, also for constrained Tikhonov regularization. In this context, we extend the concept of projected source conditions already known in Hilbert spaces to Banach spaces, and we show in the main theorem that such projected source conditions are to some extent equivalent to certain variational inequalities. The derived variational inequalities immediately yield convergence rates measured by Bregman distances.

We describe and analyze a general framework for solving ill-posed operator equations by minimizing Tikhonov-like functionals. The fitting functional may be non-metric and the operator is allowed to be non-linear and non-smooth. In comparison to former results on variational regularization with non-metric fitting functionals we significantly weaken the assumptions for proving convergence rates and, in addition, we extend the results to a wider range of rates. Two examples, coming from imaging applications, show that the developed theory is applicable to practically relevant problems.