Fixed point regularization schemes and their discretization

Nonlinear ill-posed inverse problems arise in a number of applications. One of them is a model calibration connected with the Situation that the physical law governing the process is known, but quantitative information about physical parameters is not available, and the problem is to recover the values of these parameters from noisy observations of underlying process. Usually such a problem is ill-posed and nonlinear in the Sense that the parameters depend in a discontinuous and nonlinear way an the measurements. lt is well known that the numerical treatment of nonlinear ill-posed problems requires the application of special regularization methods. The most important question in such application is the proper choice of the regularization parameter. In literature we mainly find parameter choice strategies leading to iteratively regularized methods of Gauss- Newton type. However, these theoretically well justified methods require very strong assumptions which are not always satisfied in practice. Therefore, we propose to study a new class of nonlinear regularization methods going under the narre fixed point regularization schemes. For the first time some methods from this class were discussed in the classical paper by Tikhonov and Glasko (1965), but it was done an a heuristic basis only. Based an our new approach to the regularization parameter choice we plan to develop a theory for fixed point regularization schemes, hoping that it allows to treat nonlinear illposed problems with optimal order of accuracy under much weaker nonlinearity assumptions.