| title: | Rules for choice of the regularization Parameter by solving ill-posed problems |
|---|---|
| reg no: | ETF5785 |
| project type: | Estonian Science Foundation research grant |
| subject: |
1.2. Applied Mathematics |
| status: | accepted |
| institution: | TU Faculty of Mathematics |
| head of project: | Uno Hämarik |
| duration: | 01.01.2004 - 31.12.2007 |
| description: | In this project we study ill-posed problems, i.e. problems, solution of which is unstable under data perturbations. For diminishing the influence of data perturbations the approximate solution of ill-posed problem is found by the regularization method and depends on the regularization parameter. The main question by applying regularization methods is the proper choice of the regularization parameter. In this project we concentrate namely on this choice. The proper choice of the regularization parameter according to the noise level of the data guarantees convergence of the approximate solution to the exact solution, if noise level tends to zero. It is also known that such convergence is not guaranteed for the parameter choice rules which do not use the noise level. Nevertheless large amount of recent articles is devoted to study of rules of this kind (L-curve rule, GCV-rule). The reason for popularity of these rules is that in applied ill-posed problems the noise level is not exactly known, whereas if in the classical rules for parameter choice (discrepancy principle etc) the used noise level is somewhat smaller than actual noise level is, the error of approximate solution can be arbitrarily large. Recently we succeeded to derive for parameter choice in self-adjoint problems a special rule with property, that approximate solution converges to the exact one (if noise level tends to zero) by the approximately given noise level provided that the ratio of the actual and presumed noise level is bounded. The error estimates of optimal order are proved, also. In this project we plan to continue the study of this rule for self-adjoint problems and plan to extend the results to the non-selfadjoint problems. In connection with this new strategy for the choice of the regularization parameter we hope to elaborate effective algorithms for parameter choice in case of random noise. We shall consider the case of known noise level data as well. In the recent papers we proposed for the choice of the regularization parameter the monotone error rule. In this project we plan to study the regularization properties of this rule in projection methods, in conjugate gradient method, in GMRES-method and in algorithms of A.K.Louis. We plan also to study for Lavrentiev method analogue of the monotone error rule, what we have derived recently. |
| project group | ||||
|---|---|---|---|---|
| no | name | institution | position | |
| 1. | Uno Hämarik | TU Faculty of Mathematics | Assoc.Prof. | |
| 2. | Reimo Palm | Tartu Ülikool | assistent | |
| 3. | Toomas Raus | Tartu Ülikool | dotsent vanemteadur | |