Study On Krylov Subspace Methods For Ill-posed Problems With Their Performance Analysis

With the rapid development of science and technology as well as engineering computation, the solution of linear systems of equations that arise from the discretization of linear ill-posed problems is required in numerous practical problems, such as Fredholm integral equations of the first kind, boundary value problems of inverse heat conduction equation, inverse problems of mathematical equations. The key to solving such problems is to solve the involved least-square problem. For small-scale ill-posed problems,the traditional regularization methods are often employed. However, for large-scale discrete ill-posed problems, the traditional regularization methods are often not applicable because of the limitation of computational and storage capacity.Krylov subspace methods are the most popular iterative regularization methods for solving large linear ill-posed problems. The field for solving the ill-posed problems in industrial computational models has seen an explosion of Krylov subspace methods spurred by demand due to extraordinary advances in science and engineering technology, ending with a number of toolboxes with specialized algorithms available. The characteristics revealed in numerical computation of Krylov subspace methods for dealing with these problems show the excellence of these methods when solving such related problems. For instance, they have fast convergence rate in the process of the calculation; the matrices do not need to be segmented, or even do not need to be explicitly formed. Although such kinds of methods may have semi-convergence phenomenon due to the interference of noise, the choice of appropriate regularization parameters can help to obtain a stable approximate solution. Therefore, Krylov subspace methods become a quite powerful tool to deal with this kind of problems.This paper gives an overview of the study on ill-posed problems, illustrates relevant progress of Krylov subspace methods for solving discretized ill-posed problems and frequently used types of methods, including classical methods, normal equation methods,augmented methods and variable preconditioned methods. The differences and relations among different types of algorithms as well as stopping criteria for some algorithms are analyzed. As instructed by the enriched CGLS method with the right-hand side vector and the augmented GMRES-type methods by the basis of a user-supplied subspace,a new hybrid augmented CGLS method, namely HAGCLS, is proposed by enriching CGLS simultaneously with the right-hand side vector and the basis of a user-supplied subspace. Numerical experiments for solving ill-posed problems show our HACGLS method obtains more accurate solutions in comparison with the standard CGLS method,the enriched CGLS methods respectively enriched by the right-hand side vector and the basis of a user-supplied subspace. It also indicates the feasibility and efficiency of the Krylov subspace methods of CGLS-type for solving such kind of problems.