Krylov Iteration Regularization Methods For Solving Large-scale Discrete Ill-posed Problems

The ill-posed problems are proposed in comparison with the mathematical physical equations which widely arise in physical property detection,meteorological prediction and image restoration and so on.In this paper,Krylov iteration regularization methods for solving discrete ill-posed problems are studied on the basis of existing theories and methods of the ill-posed problems.The main contents can be stated as follows: Firstly,we propose the projection fractional Tikhonov regularization method to computing an approximate solution of linear discrete ill-posed problem which combines the fractional matrices and orthogonal projection operators.Secondly,a new method for selecting orthogonal projection operator is proposed based on the deflation eigenvector,and the numerical examples show that this algorithm is better than the existing methods.Then,the Arnoldi projection fractional Tikhonov regularization method is discussed for solving large-scale discrete ill-posed problems by projecting the original problem to Krylov subspace.Furthermore,the RR-Arnoldi projection fractional Tikhonov regularization algorithm is proposed.All the proposed algorithms are programmed and the numerical examples are carried out.Numerical examples illustrate the effectiveness of the proposed methods.