Parameters-Dependent Iterative Algorithms And Constructions Of Preconditioners For Saddle Point Problems

Many of the scientific computing and engineering applicat.ions need to solve a large sparse(generalized)saddle-point type linear system of equations,such as computational fluid dynamics,constraints and weighted least squares estimate and constrained optimiza-tion and so forth.Therefore,the solution of(generalized)saddle point problems has become a hot international research topic in recent decades.In the field of scientific com-puting,the famous method for solving the general large sparse linear system of equations is iterative method.Iterative methods used to solve linear system of equations mainly in-clude stationary iterative method based on matrix splitting and Krylov subspace method based on projection process.As is known to all,there is not an universal method for solving all linear systems of equations.That is to say,some methods applied to a problem may not be applicable to another one.The selection of solution method usually depends on structures and properties of the coefficient matrix of linear system of equations.For different application backgrounds,furthermore,the coefficient matrix of linear system of equations often has different properties and structures.In this dissertation,we focus on detecting several classes of large sparse saddle point problems with special structures and properties,include nonsingular saddle point prob-lems,singular saddle point problems and generalized saddle point problems which are e-quivalent with complex symmetric linear systems of equations.We present several efficient iterative algorithms and preconditioners and analyse the(semi-)convergent properties of the corresponding iterative methods and give some numerical experiments.The main achievements of this dissertation are as follows:In Chapter 2,for nonsingular saddle point problems,we first extend the HSS-based sequential two-stage method for solving non-Hermitian saddle point problems and analyse the convergence of the generalized method and the property of spectral radius of the iter-ative matrix.Numerical results demonstrate that the generalized method can be used to solve the non-Hermitian nonsingular saddle point problems with(1,1)-block being Hermi-tian dominant or non-Hermitian dominant while the original method is unfit for solving the ones with(1,1)-block being weak Hermitian dominant or skew-Hermitian dominant.Secondly,we propose the generalized shift-splitting iterative method and preconditioner and analyse the unconditionly convergent property of the generalized shift-splitting iter-ative method.Numerical results show that the generalized shift-splitting preconditioner is efficient.Meanwhile,we compare the Anderson acceleration for the generalized shift-splitting iterative method with the corresponding left preconditioned restarted GMRES method.Finally,we present the modified generalized preconditioned parameterized in-exact Uzawa(MGPPIU)method and derive the sufficient conditions for its convergence.Numerical results show that MGPPIU method can be better than GPPIU,PPIU and PIU methods in some situation.In Chapter 3,for singular saddle point problems,we first propose the generalized shift-splitting iterative method and preconditioner and analyse the unconditionly semi-convergent property of the generalized shift-splitting iterative method.Numerical results show that the generalized shift-splitting preconditioner is efficient.At the same time,we also compare the numerical performances of Anderson acceleration for the generalized shift-splitting iterative method with the corresponding left preconditioned restarted GM-RES method.Then,we present the MGPPIU method and derive the sufficient conditions for its semi-convergence.Numerical results show that MGPPIU method is superior to GPPIU,PPIU and PIU methods in some cases.In Chapter 4,for the real equivalent form of complex symmetric linear system of equa-tions,we first present the AOR-Uzawa iterative method and preconditioner and analyze the convergence of AOR-Uzawa iterative method and the spectral property of the precon-ditioned matrix.Numerical results demonstrate that the AOR-Uzawa iterative method and preconditioner are efficient.Secondly,we propose the sequential two-stage method and detect its convergent property and the property of spectral radius of the iterative matrix.Numerical results show that the sequential two-stage method is feasible.Finally,we present the Anderson accelerations of the GSOR and PGSOR methods.Numerical ex-periments show the numerical performances of the Anderson acceleration and we compare the Anderson acceleration with the corresponding left preconditioned restarted GMRES method.