Discretization strategy for linear ill-posed problems in variable Hilbert scales

The authors study the regularization of projection methods for solving linear ill-posed 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.