Optimality for ill-posed problems under general source conditions

• Published in: Numerical functional analysis and optimization Vol. 19; no. 3-4; pp. 377 - 398
• Main Author: Tautenhahn, Ulrich
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Marcel Dekker, Inc 01.01.1998; Taylor & Francis
• Subjects: 1991 Mathematics Subject Classifications:65M30; 1991.Mathematics Subject Classifications 35R25; Exact sciences and technology; Ill-posed problems; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; optimal error bounds; optimal regularization methods; Partial differential equations; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Sciences and techniques of general use; Regularization method; Spectral method; Error estimation; Optimal error estimation; Optimality; Best approximation; Optimal approximation; Hilbert space; Ill posed problem; Spectral method optimality
• ISSN: 0163-0563; 1532-2467
• DOI: 10.1080/01630569808816834

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.