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

The ill-posed problems are the hot issues in the science field nowadays. The problems arewell-posed problems if its solutions are existent, unique and stable. If at least one condition is notmet, it is ill-posed. However, the greatest difficulty about the solution of ill-posed problems lies inthe instability of the solution. That the small errors in practical measurement lead to seriousdeviation between approximate solutions and exact solutions results in the difficulty in solvingill-posed problems. Strictly speaking, the ill-posed problems are infinite dimensional. Almost allof ill-posed problems must be discreted and converted into finite dimensional from the aspect ofnumerical implementation. This paper studies the numerical algorithms of linear ill-posed discreteproblems. Classical regularization methods are efficient for many ill-posed problems. However,when classical regularization methods are used in large scale problems, they are inferior toiterative regularization methods. This paper mainly focuses on the RRGMRES method, the rangerestricted method, which is one of the iterative regularization methods. Like most iterativeregularization methods, it possesses the properties of semi-convergence, that is, as the iterativenumber increases, the relative error between the regularization solution and the exact solutiondecreases to the minimum and then increases rapidly. It is exactly the properties ofsemi-convergence that leads to extremely poor result of the regularization RRGMRES methodwithout proper stop criterion. Firstly, the paper proposes a new method to confirm thetriangle-condition L-curve parameter of the regularization RRGMRES method though analyzingthe regularization properties of the RRGMRES. Secondly, as vector extrapolation can acceleratethe speed of the iteration process, the paper proposes another new method by connecting vectorextrapolation with the RRGMRES method so as to confirm the regularization parameter. At last,this paper conducts numerous classical numerical experiments and comparison of the improvedalgorithms. Numerical experiments indicate that the two new improved regularization methods arefeasible and efficient.