The Improved Tikhonov Algorithms For Solving Linear Discrete Ill-posed Problems

With the development of the technology and the improvement of productivity, the ill-posed problems widely arise in many areas such as geophysics, automatic control and so on. Regularization methods exist for computing the stable solution approximations for ill-posed problems. In this thesis, regularization methods for solving linear discrete ill-posed problems are considered. Firstly, based on the two popular methods for solving linear discrete ill-posed problems, the Tikhonov regularization method and the TSVD regularization method, a hybrid Tikhonov regularization method for linear discrete ill-posed problems is proposed. Secondly, an Arnoldi-Fractional Tikhonov regularization method for large scale linear discrete ill-posed problems is presented via applying the Fractional Tikhonov regularization to the projection algorithm. Further more, the generalized Arnoldi-Fractional Tikhonov method and the range restricted Arnoldi-Fractional Tikhonov method are proposed in the follows. At last, this thesis conducts numerous classical numerical experiments on the improved methods proposed above. Numerical experiments and comparisons indicate that the new improved regularization methods are feasible and efficient.