Research On Mixed-parameters Class SOR Iteration Method For Saddle Point Problems

The saddle point problem comes from many engineering calculation fields,such as constrained optimization calculation,fluid mechanics,electromagnetic calculation,financial engineering,etc.Under appropriate boundary conditions,these problems can be transformed into saddle point problems for solving large sparse linear systems by discretization methods.The solution is a key step in engineering calculation.Therefore,how to design efficient,robust and practical numerical algorithms according to the specific saddle point structure has become a research hotspot in the field of scientific computing.Based on the current research progress,this paper studies the iterative methods of two kinds of saddle point systems,and proposes several new numerical methods.The main research work is as follows:1.For the asymmetric saddle point problem,the defects of the PSS-GSOR iterative method in the relevant literature are analyzed,and a modified PSS-GSOR(MPSS-GSOR)iterative method is constructed to further improve the efficiency of the iterative method.By introducing new parameters to improve the sensitivity of the iterative correlation coefficient matrix,and using the matrix eigenvalue theory,the convergence of the given iteration is proved,and the range of relevant parameters is obtained.Compared with the PSS-GSOR iteration method,the numerical results show that the MPSS-GSOR iteration method has faster convergence rate.2.For the asymmetric saddle point problem,the defects of the MSOR-Like iterative method in the relevant literature are analyzed,and an accelerated MSOR-Like(AMSORLike)and a modified MSOR-Like(MMSOR-Like)iterative method are constructed to further improve the efficiency of the iterative method.By introducing new parameters to improve the sensitivity of the iterative correlation coefficient matrix,and using the matrix eigenvalue theory,the convergence of these two new iterative methods is proved,and the range of relevant parameters is obtained.3.According to the MSOR-Like method for nonsingular saddle point problems,a new GMSOR-Like method for generalized saddle point problems is proposed.After splitting the coefficient matrix,the sensitivity of the iterative correlation coefficient matrix is improved by introducing a positive definite matrix and two parameters.At the same time,the convergence of the given iteration is proved by using the matrix eigenvalue theory,and the range of relevant parameters is obtained.Numerical experiments show that the GMSOR-Like method is used to solve the generalized saddle point problem,which has significantly improved convergence efficiency compared with similar iterative methods.