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In this paper we prove that the Landweber iteration is a stable method for solving nonlinear ill-posed problems. For perturbed data with noise level δ we propose a stopping rule that yields the convergence rate O(δ 1/2 ) under appropriate conditions. We illustrate these conditions for a few examples.Mathematics Subject Classification (1991): 65J15, 65J20, 47H17

There exists a vast literature on convergence rates results for Tikhonov regularized minimizers. We are concerned with the solution of nonlinear ill-posed operator equations. The first convergence rates results for such problems have been developed by Engl, Kunisch and Neubauer in 1989. While these results apply for operator equations formulated in Hilbert spaces, the results of Burger and Osher from 2004, more generally, apply to operators formulated in Banach spaces. Recently, Resmerita et al. presented a modification of the convergence rates result of Burger and Osher which turns out a complete generalization of the rates result of Engl et. al. In all these papers relatively strong regularity assumptions are made. However, it has been observed numerically, that violations of the smoothness assumptions of the operator do not necessarily affect the convergence rate negatively. We take this observation and weaken the smoothness assumptions on the operator and prove a novel convergence rate result. The most significant difference in this result to the previous ones is that the source condition is formulated as a variational inequality and not as an equation as before. As examples we present a phase retrieval problem and a specific inverse option pricing problem, both studied in the literature before. For the inverse finance problem, the new approach allows us to bridge the gap to a singular case, where the operator smoothness degenerates just when the degree of ill-posedness is minimal.

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Motivated by the theoretical and practical results in compressed sensing, efforts have been undertaken by the inverse problems community to derive analogous results, for instance linear convergence rates, for Tikhonov regularization with 1 -penalty term for the solution of ill-posed equations. Conceptually, the main difference between these two fields is that regularization in general is an unconstrained optimization problem, while in compressed sensing a constrained one is used. Since the two methods have been developed in two different communities, the theoretical approaches to them appear to be rather different: In compressed sensing, the restricted isometry property seems to be central for proving linear convergence rates, whereas in regularization theory range or source conditions are imposed. The paper gives a common meaning to the seemingly different conditions and puts them into perspective with the conditions from the respective other community. A particularly important observation is that the range condition together with an injectivity condition is weaker than the restricted isometry property. Under the weaker conditions, linear convergence rates can be proven for compressed sensing and for Tikhonov regularization. Thus existing results from the literature can be improved based on a unified analysis. In particular, the range condition is shown to be the weakest possible condition that permits the derivation of linear convergence rates for Tikhonov regularization with a priori parameter choice.

We consider the stable approximation of sparse solutions to nonlinear operator equations by means of Tikhonov regularization with a subquadratic penalty term. Imposing certain assumptions, which for a linear operator are equivalent to the standard range condition, we derive the usual convergence rate O( √ δ) of the regularized solutions in dependence of the noise level δ. Particular emphasis lies on the case, where the true solution is known to have a sparse representation in a given basis. In this case, if the differential of the operator satisfies a certain injectivity condition, we can show that the actual convergence rate improves up to O(δ).

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