Our research group considers ill-posed problems with different cases of information about the noise level of the data:
- there is no information about the noise level;
- rough estimate of the noise level is given;
- exact noise level is given.
We investigate solving of ill-posed problems mainly by the following methods:
- Tikhonov and Lavrentiev methods and their iterated and extrapolated versions;
- iteration methods (Landweber, conjugate gradient type methods etc);
- projection methods (least error method, least-squres method, collocation method).
Main topic of our group is the proper choice of the regularization parameter (the number of iterations in iteration methods, the dimension of subspace in projection methods). In cases 2, 3 of the noise level information we study choices of the regularization parameter, which guarantee the convergence of the approximate solution to the exact solution, if noise level tends to zero. For case 1 (no noise level information) we propose heuristic rules, which give good parameters in most cases.
We have published papers on the following research topics.
- General theory of ill-posed problems
- Self-regularization of ill-posed problems by projection methods
- Extrapolation of Tikhonov and Lavrentiev methods
- Optimality and quasioptimality of the choice of the regularization parameter
- Heuristic rules for choice of the regularization parameter
- Choice of the regularization parameter in the case of inexact noise level of the data
- Choice of the regularization parameter by the monotone error rule
- Choice of the regularization parameter by (modified) discrepancy principle and by balancing principle
- Regularized projection methods
- Analysis of specific ill-posed problems