The Construction Of Preconditioners For Generalized Saddle Point Problems And Complex Linear Systems

With the progress and development of science,many scientific computing problems will eventually become the problem of solving a large linear system.The saddle point problem is one of these problems.In engineering and scientific calculation,the solution of the saddle point problem has a wide range of applications and has certain research value,such as computational fluid dynamics,image reconstruction and registration,elliptical partial differential equation mixed finite element problem weighted least square problem time harmonic vortex model and so on.The coefficient matrix of the saddle point problem generally has two characteristics:positive or negative eigenvalues,strong uncertainty;diagonal elements are not dominant,do not have diagonal dominance.Because of these two characteristics,in some cases,the direct method may not be as efficient as the iterative method.If the iterative method is directly used to solve the saddle point problem,the iterative speed may not be very ideal and sometimes even does not converge,especially when the coefficient matrix is a large sparse matrix,which indicates that preconditioning technology is needed to solve some saddle point problems,which makes the preconditioned linear system have better properties.In this paper,we discuss a class of common generalized saddle point problems and a class of complex linear systems(which can be transformed into generalized saddle point problems equivalently),construct their respective preconditioners,and accelerate Krylov subspace method.The main tasks are as follows:1.Aiming at the generalized saddle point problem,based on SHSS preconditioner,the residual matrix and residual equation are analyzed.By adjusting the division of matrix multiplication,two new types of preconditioners are constructed by adding parameter matrix,and the corresponding iterative method,the expression of optimal parameters,as well as the spectral properties of preconditioned matrix are given.Finally,the validity and feasibility of the two new preconditioners are proved by experimental data and images.2.In order to solve the special saddle point problem generated by the equivalent transformation of the complex linear system,a new preconditioner is constructed based on the MB preconditioner,and the preconditioned matrix is analyzed:the characteristic value analysis,the feature vector analysis and the proof of the minimum polynomial theorem.The effectiveness and feasibility of this new type of preconditioner are described with experimental data and images.