Geometry of linear ill-posed problems in variable Hilbert scales

The authors study the best possible accuracy of recovering the solution from linear ill-posed problems in variable Hilbert scales. A priori smoothness of the solution is expressed in terms of general source conditions, given through index functions. The 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.