Error Estimates for Regularization Methods in Hilbert Scales

In this paper we study regularization methods to reconstruct the solution x*of the linear ill-posed problem Ax = y, A : X → Y, from noisy data yδ∈ Y. The regularization methods are of the general form$x^\delta_\alpha = \bar x + g_\alpha (B^{-s} A^\ast A) B^{-s} A^\ast (y^\delta - A\bar x)$where B denotes an unbounded self-adjoint strictly positive definite operator in the Hilbert space X. Assuming ∥ Ax∥ ∼ ∥ x∥-aand$\parallel x^\ast - \bar x\parallel_p \leq\le E$for some$\bar x\in X, a \geq 0$and p ≥ 0 (∥ x∥r= ∥ Br/2x∥ is the norm in a Hilbert scale (Xr)r ∈ R) we derive error estimates which show that the accuracy of the above regularization methods is order optimal provided that the function gα(λ), the regularization parameter α, and the parameter s are chosen properly.