Research On The Numerical Methods Of Saddle Point Problems

The saddle point problem which is a special linear system has become a hot topic in recent years.Due to its broad applications,saddle point problems have been studied by many scholars.Moreover,lot of theories and efficient numerical methods have been presented.In this thesis,we mainly discuss some numerical algorithms for solving saddle point problems.Some improvements of the existed numerical methods are obtained and some efficient iterative methods and preconditioners are proposed.The outline of the thesis is as follows:In Chapter 1,some basic results of saddle point problems including the background,theory and numerical methods are reviewed.Some definitions,preliminaries and basic splitting methods are also given,which will be used in our later discussion.In Chapter 2,we analyse the semi-convergence property of the ULT iterative method for solving the singular saddle point problems and study some properties of eigenvalues of the corresponding iteration matrix.Moreover,pseudo-optimal iteration parameters and the corresponding pseudo-optimal semi-convergence factor of the ULT method for saddle point problems are presented.Numerical experiments illustrate the feasibility and efficiency of the ULT method to solve large sparse singular saddle point problems.In Chapter 3,we proposed the Extended shift-splitting(ESS)preconditioner for saddle point problems.Moreover,we introduce some special ESS preconditioners.ES-S preconditioner is an generalization of the shift-splitting(SS)preconditioner.Theory analysis and numerical tests show that the ESS method outperforms the SS method for solving saddle point problems.In Chapter 4,based on a parameterized splitting of the coefficient matrix of the saddle point problem,a fixed iteration was proposed.Moreover,some parameterized matrix splittings(PMS preconditioners)are obtained.We discuss the properties about the eigenvalues and eigenvectors of the iteration matrix of the PMS iterative method.For the corresponding preconditioned matrix,we analysed its spectral property.Numerical experiments show the efficiency of this method.In Chapter 5,we study another efficient upper and lower triangular splitting pre-conditioner to solve large sparse saddle point problems.Since we introduce a parameter?,we call this kind methods parameterized upper and lower triangular(PULT)splitting methods.Theoretical analysis and numerical experiments show that the improvement has good theoretical significance and numerical effect.In Chapter 6,we discuss a shift-splitting preconditioner for a class of block two-by-two linear systems.This kind of linear systems are special generalized saddle point problems.The proposed preconditioner is extracted from a stationary iterative method which is unconditionally convergent.Moreover,the eigenvalue distribution of the corre-sponding preconditioned matrix is studied.Numerical experiments are presented to show that our new preconditioner can be quite competitive when used to precondition Krylov subspace iterative methods such as GMRES.In Chapter 7,we still consider the block two-by-two linear system studied in Chapter 6.Our main contribution is accelerating the convergence of the PIU algorithm by making use of the extrapolation technique which is based on eigenvalues of the iteration matrix.These accelerated parameterized inexact Uzawa algorithms involve two iteration param-eters whose special choices can recover the parameterized inexact Uzawa algorithms,as well as yield new ones.First,the accelerated model for the PIU algorithm is established and the accelerated PIU algorithm is presented.Then we study the convergence property of the corrected PIU algorithm.In addition,we present the optimal iteration parameter and the corresponding optimal convergence factor for the PIU method.Numerical exper-iments are presented to illustrate the correctness of our theoretical results and examine the numerical effectiveness of the new method.Some concluding remarks are given in Chapter 8,and our future researches are also presented here.