Research On Splitting Iterative Methods And Preconditioning Techniques For Linear System Of Equations

Solutions of large-scale sparse linear systems arise widely in scientific engineering computing.Moreover,research on solving method of large-scale sparse systems of linear algebraic equations has become one of the core problems in large-scale science and engineering computation,and such research show great theoretical and practical significance.In this doctoral dissertation,some splitting iterative methods for solving large sparse linear systems of linear algebraic equations are studied deeply.In particular,convergence properties of matrix splitting methods for solving some linear systems have been analyzed and discussed in terms of fractional diffusion equations,shifted linear system,saddle point problem and linear complementarity problems.This dissertation consists of seven chapters,which are mainly divided into five parts as follows:Part 1 contributes to solve complex symmetric system by generalizing modified Hermitian and skew-Hermitian splitting iterative method.First,we propose the generalized MHSS(GMHSS)method which is the extension of MHSS splitting method.Using this new method,we establish convergence theorems of GMHSS iterative method for the new method.Lastly,numerical experiments are carried out and experimental results show that the new iterative method is effective.Part 2 is devoted to investigate the implicit finite difference scheme with the shifted Gr¨unwald formula to discretize the fractional diffusion equations with constant coefficients.Since the coefficient matrix of resulting system possesses the structure of positive definite Toeplitz-like,then we use the Hermitian and skew-Hermitian splitting method for solving the Toeplitz-like system.Two linear subsystems are needed to solved using the Hermitian and skew-Hermitian splitting iterative method.Krylov subspace methods are proposed to solve each subsystem via using the fast Fourier transforms(FFTs),which can reduce the computational cost of the matrix-vector multiplication.At the same time,we can use some circulant preconditioners,such as Strang's circulant preconditioner and the T.Chan's preconditioner,to accelerate the convergence rate of Krylov subspace methods.Moreover,we present the convergence analysis and prove the spectrum properties of the preconditioned matrices to be clustered around 1.Then the superlinear convergence rates of the proposed iterative algorithms are obtained.Part 3 is to discuss the problem of updating preconditioning technique for solving the shifted linear systems and a new preconditioner is proposed to update the precondition matrix.Based on the construction of preconditioners by modifying the factorization of matrix ,the new preconditioners are obtained for different .Then we discuss the properties of proposed preconditioners and the bounds of the eigenvalues for the preconditioned matrices.The proposed preconditioners generalize the updated technique in literature [1].Numerical experiments show that the new updating technique is effective when the parameter lies in a relatively large range.Based on modulus-based matrix splitting iteration method,Part 4 is to discuss how to accelerate the modulus-based matrix splitting iteration method.The modulus-based matrix splitting iteration and its relaxed variants are employed as inner iterations for solving linear systems approximately,and the iterative process of the new method is presented in detail.Particularly,we provide convergence results and properties for the proposed method when the system matrices are positive-definite matrices and +-matrices.Numerical experiments show that the proposed method is more efficient than the modulusbased matrix splitting iteration method[2]in items of iteration steps and CPU under some conditions.Part 5 discusses about the solution of saddle point problem.First,a generalized Uzawa-SOR(GUSOR)method for solving saddle point problem is established,which is the extension of the USOR method[3].Then we analyze some properties of eigenvalue and eigenvector,and the convergence of the generalized Uzawa-SOR is discussed under suitable restrictions on the iteration parameters.Lastly,numerical experiments are carried out and experimental results show that our proposed method with appropriate parameters has faster convergence rate than that of the USOR method[3].