Research On Improved Uzawa Class Iterative Algorithm For Solving Saddle Point Problems

In recent years,the saddle point problem has been widely used in many fields of scientific engineering such as computational electromagnetics,high-order differential equations,optimization problems,computational fluid dynamics and so on.Although the development of computer science is more and more comprehensive,the solution to the saddle-point problem still faces the shortcomings of large storage space,complicated computation and long time-consuming.Therefore,it is the key problem that the saddle point problem needs to be solved to establish the algorithm with small computation and accurate result.Starting from solving the classical saddle point problem,this article firstly describes the research background and current research situation,and presents the iterative schemes for HSS,Uzawa and SOR methods.The iterative format and convergence analysis of Uzawa-AOR method and its modified algorithm are introduced in detail.It can be seen that the Uzawa-AOR method and the modified Uzawa-AOR method both are an inner-outer two-layer iterative method combining the Uzawa method with the AOR method.On this basis,a new type of bidirectional symmetric modified Uzawa method is constructed and the corresponding theoretical analysis and numerical simulation experiments are given.The convergence theorem shows that by choosing the appropriate parameters s,t,the range of the parameters ? in the original method is expanded.Therefore,the new method is more applicable than other methods.The Uzawa-PSS method is an iterative method for solving the non-Hermitian saddle point problem using positive definite and skew-Hermitian splitting and Uzawa iterative methods.PSS iterative method is the extension of HSS iterative method,mainly used to solve positive saddle point problem.Every step in PSS iterative method needs to solve two linear subsystems,al+P and al+S.The solution of linear subsystems al+P can be directly obtained by CG method,but the linear subsystems ?I+S are relatively difficult to solve.So in order to overcome this shortcoming,single-step PSS iterative method is used to approximate xk+1,so an improved Uzawa-PSS iterative method is constructed.The convergence of the nonsingular saddle point problem and the semiconvergence of the singular saddle point problem with the new method are analyzed in detail and the numerical simulation experiments are carried out.By comparison,it can be concluded that the Uzawa-PSS iterative method,which has been improved in terms of time cost and number of iterations,has obvious advantages.