AN OPTIMAL ORDER REGULARIZATION METHOD WHICH DOES NOT USE ADDITIONAL SMOOTHNESS ASSUMPTIONS

This paper defines an optimal method to reconstruct the solutions of operator equations of the first kind. Only the case of compact operators is considered. The method is in principle a discrepancy method. It does not require any additional knowledge about the solution and is optimal for all standard smoothness assumptions. In order to analyze the properties of the new regularization method variable Hilbert scales are introduced and several well-known results for Hilbert scales are generalized. Convergence theorems for classes of optimal and suboptimal methods are derived from a generalization of the interpolation inequality.