The Design And Application Of Regularization Computational Methods For Ill-posed Problems

The ill-posed problems arise from the practical problems in physics, biology, medicine and geography, etc. Large scaled linear systems arising from discretization of the practical problems are highly ill-posed, so the high performance regularization numeric solving process is needed. This paper focuses on solving the exceedingly ill-posed matrix equation generated from the practical problems by establishing an efficient regularization method.We introduce the mathematical theory of ill-posed problems, regularization theory for solving ill-posed problems, typical regularization methods and selection of regularization parameter. Tikhonov regularization method is widely used in certain inverse problems, in which the regularization term that is introduced to lessen the ill-posedness in inverse problems can be seen as containing prior information of the problem. We can change the prior information contained in the regularization term by choosing different regularization operator. In order to compute stable solutions and reduce multi-solution, on the one hand, this paper presents an improved Tikhonov regularization method; on the other hand, this paper designs a kind of improved two-parameter regularization method based on Tikhonov regularization method, which introduce a regularization term with second order regularization operator. Regularization parameter is the key of achieving the regularization algorithm, but the selection of the optimal regularization parameter has been no unified and easy to implement method. Morozov's discrepancy principle, L-curve criterion and generalized cross-validation are applied to determine the optimal value of the regularization parameter. To show the superiority and validity of the improved Tikhonov regularization and the improved two-parameter regularization method, it is carried out to the theoretical mode, and numerical performance indicates that the two proposed methods are not only high precision, but also stability for random noises of the data.Finally, we implement the two proposed methods to a practical mode of electrical conductivity imaging, and the clearer imaging result reveals that the solution is excellent. Electrical conductivity imaging consists with the geological features.