The authors study the best possible accuracy of recovering the unknown solution from linear ill-posed problems in variable Hilbert scales. A priori smoothness of the unknown solution is expressed in terms of general source conditions, given through index functions. Emphasis is on geometric concepts. The notion of regularization is appropriately generalized, and the interplay between qualification of regularization and index function becomes visible. A general adaptation strategy is presented and its optimality properties are studied.

In this paper we discuss a relation between Learning Theory and Regularization of linear ill-posed inverse problems. It is well known that Tikhonov regularization can be profitably used in the context of supervised learning, where it usually goes under the name of regularized least-squares algorithm. Moreover, the gradient descent algorithm was studied recently, which is an analog of Landweber regularization scheme. In this paper we show that a notion of regularization defined according to what is usually done for ill-posed inverse problems allows to derive learning algorithms which are consistent and provide a fast convergence rate. It turns out that for priors expressed in term of variable Hilbert scales in reproducing kernel Hilbert spaces our results for Tikhonov regularization match those in Smale and Zhou [Learning theory estimates via integral operators and their approximations, submitted for publication, retrievable at http://www.tti-c.org/smale.html , 2005] and improve the results for Landweber iterations obtained in Yao et al. [On early stopping in gradient descent learning, Constructive Approximation (2005), submitted for publication]. The remarkable fact is that our analysis shows that the same properties are shared by a large class of learning algorithms which are essentially all the linear regularization schemes. The concept of operator monotone functions turns out to be an important tool for the analysis.

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The authors study the regularization of projection methods for solving linear illposed problems with compact and injective linear operators in Hilbert spaces. The smoothness of the unknown solution is given in terms of general source conditions, such that the framework of variable Hilbert scales is suitable.The structure of the error is analysed in terms of the noise level, the regularization parameter and as a function of other parameters, driving the discretization. As a result, a strategy is proposed which automatically adapts to the unknown source condition, uniformly for certain classes, and provides the optimal order of accuracy.