Optimality for ill-posed problems under general source conditions

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 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 (λ) serve as model examples.

Keywords

• Ill-posed problems,
• optimal error bounds,
• optimal regularization methods,
• 1991 Mathematics Subject Classifications:65M30,
• 1991.Mathematics Subject Classifications 35R25