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.

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.

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Abstract. The authors study parameter choice strategies for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces. The effectiveness of any parameter choice for obtaining convergence rates depends on the interplay of the solution smoothness and the nonlinearity structure, and it can be expressed concisely in terms of variational inequalities. Such inequalities are link conditions between the penalty term, the norm misfit and the corresponding error measure. The parameter choices under consideration include an a priori choice, the discrepancy principle as well as the Lepskiȋ principle. For the convenience of the reader the authors review in an appendix a few instances where the validity of a variational inequality can be established.