Optimality for ill-posed problems under general source conditions - Tautenhahn, Ulrich (Numerical functional analysis and optimization, 1998)
In this paper we consider linear ill-posed problems where instead of y noisy data y δ are available with and is a linear operator between Hilbert spaces X and Y. Assuming the general source condition with appropriate functions φ we study following questions:(i) which (best possible) accuracy can be obtained for identifying x from under the assumptions (ii) are there special regularization methods which guarantee this best possible accuracy, i.e., which are optimal on the set M δ,E ? Concerning question (i) we prove that under certain conditions there holds inf sup with where the 'inf' is taken over all methods and the 'sup' is taken over all .and Concerning question (ii) we prove the optimality of a general class of regularization methods and specify our general optimality results to Tikhonov type methods and to spectral methods. Heat equation problems backward in time which are characterized by different functions φ(λ) serve as model examples.