Plenary Lecture
Stable Solution of Ill-Posed Problems by Projection Methods
Professor Uno Hämarik
Institute of Mathematics
University of Tartu
Estonia
E-mail: Uno.Hamarik@ut.ee
Abstract: Ill-posed problems are problems, solutions of which are unstable under data perturbations (which may be for example the measurement errors). Typical examples of the ill-posed problems are integral equations of the first kind. For the solution of ill-posed problems special regularization methods are derived (Tikhonov method, iterative methods etc). For numerical realization of these methods on the computer their discretization is unavoidable. In some cases successful discretization method can be viewed as regularization method and then other regularization is not needed. Namely, if the discretization method converges in case of exact data, then in case of noisy data we can choose the discretization step in dependence of the noise level of the data in such a way, that the solution of the discretized problem converges to the solution of continuous problem if noise level tends to zero. Such phenomenon is called self-regularization. We consider self-regularization of ill-posed problems in Hilbert and Banach spaces by the following projection methods: least squares method, least error method, collocation method. In these methods the regularization parameter is the dimension of the projected equation. We choose this dimension by the discrepancy principle or by the monotone error rule and give convergence results.
Brief Biography of the Speaker: Uno Hämarik is the associated professor of the Institute of Mathematics of the Universityof Tartu, Estonia. He is born in 1955, obtained Dipl.-Math. degree from the University of Tartu, in 1978 and PhD from the Institute of Mathematics and Mechanics of Ural Centre of UdSSR, Sverdlovsk in 1986. He has worked at the University of Tartu 1978-1981 as Assistant Prof., 1984-1995 as researcher and senior researcher and from 1995 as the Assoc. Professor. U. Hämarik has held several appointments as Visiting Professor in Germany (in Universities of Kaiserslautern, Zittau/Görlitz and Kiel). He is a member of the Editorial Boards of the journals "Abstract and Applied Analysis" and "Mathematical Modelling and Analysis" and served as a referee for 22 international journals (SIAM J. Numerical Analysis, SIAM J. Optimization and Control, Inverse Problems, Numerical Functional Analysis and Optimization etc). U. Hämarik has published more than 50 papers in research journals and conference proceedings. Main topic has been regularization of ill-posed problems, especially self-regularization of ill-posed problems by projection methods and rules for the choice of the regularization parameter in many regularization methods (Tikhonov method, iterative methods, projection methods).