Researches On Iterative Algorithms For Linear Systems With Special Block Structures

In many important fields of scientific and engineering,such as digital image processing,computational fluid dynamics,structural dynamics,reservoir modeling,electromagnetic problems and constrained optimization problems,yield large-scale sparse linear systems with different block structures by using appropriate discretization.Numerical solution of such system of linear equations becomes one of the core issues of scientific and engineering computing,which has great theoretical significance and practical application value.This dissertation focus on numerical solutions of several kinds of large sparse linear systems with special block structures,by utilizing the block structure or properties of the coefficient matrix,a series of iterative methods and preconditioners are proposed.The main achievements are as follows:For nonsingular saddle point problems,we study two types of effective iterative methods and preconditioners.Firstly,we present an accelerated symmetric SORlike(ASSOR)iterative method in real domain and analyze the convergence of the ASSOR method under suitable restrictions.Numerical examples show that the ASSOR is an efficient iterative method.Secondly,we establish a class of Uzawa positive definite and positive semi-definite splitting(Uzawa-PPS)iterative method and preconditioner in complex domain.The convergence properties of the proposed method and the spectral properties of the preconditioned matrix are discussed.Numerical experiments are given to confirm the feasibility and effectiveness of the Uzawa-PPS method and preconditioner.For nonsingular complex linear systems,we study three classes of effective Euler-extrapolated iterative methods and preconditioners.Firstly,we propose an Euler-extrapolated Hermitian/skew-Hermitian splitting(E-HS)iterative method and preconditioner with symmetric positive semi-definite sub-block.The convergence properties and optimal iteration parameter of the E-HS method are presented,and the eigenvalues distribution of the preconditioned matrix are studied.Secondly,we construct a regularized E-HS(RE-HS)iterative method and preconditioner under the same assumption.The convergence properties of the RE-HS method and the spectral properties of the preconditioned matrix are studied.Thirdly,we propose an alternating E-HS(AE-HS)iterative method and preconditioner with positive definite sub-block.Theoretical analyses show that the AE-HS method is unconditionally convergent.Moreover,numerical results are given to show the feasibility and effectiveness of the corresponding Euler-extrapolated iterative methods and preconditioners.For nonsingular complex linear systems,we study two types of effective singlestep iterative methods and preconditioners.Firstly,we consider the parameterized single-step HSS(P-SHSS)iterative method and preconditioner and analyze the semiconvergence properties of P-SHSS method.The spectral properties of the preconditioned matrix and the quasi-optimal iteration parameters are discussed in detail.Secondly,we consider the RE-HS iterative method and preconditioner and analyze the semi-convergence properties of RE-HS method.We study the optimal iteration parameters of RE-HS method and the spectral properties of the preconditioned matrix.Numerical results are given to confirm the feasibility and effectiveness of P-SHSS and RE-HS methods either as solvers or as preconditioners.For nonsingular block two-by-two linear systems,we study three kinds of effective Givens-extrapolated iterative methods and preconditioners.Firstly,we propose a kind of Givens-extrapolated block splitting iterative methods and preconditioners with symmetric positive semi-definite sub-block.The convergence properties and optimal iteration parameter of the proposed method are discussed,and the eigen-properties of the preconditioned matrix are studied.Secondly,we discuss a class of inexact Givens-extrapolated block splitting preconditioner and analyze the spectral properties of the preconditioned matrix.Thirdly,we study the Givensextrapolated SSOR(G-SSOR)iterative method.We analyze the convergence properties of G-SSOR method and optimal iteration parameters.Moreover,numerical results are presented to show the feasibility and effectiveness of the corresponding Givens-extrapolated iterative methods and preconditioners.