在科学计算及工程应用中经常遇到复对称线性系统问题,近年来对一种特殊类型的复对称线性系统的研究已成为一个热点.基于白中治等的PMHSS方法(Bai Z Z,Benzi M,Chen F,Wang Z Q.Preconditioned MHSS iteration methods for a class of bl...在科学计算及工程应用中经常遇到复对称线性系统问题,近年来对一种特殊类型的复对称线性系统的研究已成为一个热点.基于白中治等的PMHSS方法(Bai Z Z,Benzi M,Chen F,Wang Z Q.Preconditioned MHSS iteration methods for a class of block twoby-two linear systems with applications to distributed control problems.IMA J Numer Anal,2013,33:343-369),提出一类新的PMHSS迭代法用于求解这种特殊形式的复对称线性系统,给出新方法的收敛性理论以及最优参数的表达式,最后用数值例子展示了新方法的有效性.展开更多
The modified Hermitian and skew-Hermitian splitting （MHSS） iteration method and preconditioned MHSS （PMHSS） iteration method were introduced respectively. In the paper, on the basis of the MHSS iteration method, w...The modified Hermitian and skew-Hermitian splitting （MHSS） iteration method and preconditioned MHSS （PMHSS） iteration method were introduced respectively. In the paper, on the basis of the MHSS iteration method, we present a PMHSS iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and complex symmetric positive definite/semi-definite matrices. Under suitable conditions, we prove the convergence of the PMHSS iteration method and discuss the spectral properties of the preconditioned matrix. Moreover, to reduce the computing cost, we establish an inexact variant of the PMHSS iteration method and analyze its convergence property in detail. Numerical results show that the PMHSS iteration method and its inexact variant are efficient and robust solvers for this class of continuous Sylvester equations.展开更多
Based on the PMHSS preconditioning matrix, we construct a class of rotated block triangular preconditioners for block two-by-two matrices of real square blocks, and analyze the eigen-properties of the corresponding pr...Based on the PMHSS preconditioning matrix, we construct a class of rotated block triangular preconditioners for block two-by-two matrices of real square blocks, and analyze the eigen-properties of the corresponding preconditioned matrices. Numerical experiments show that these rotated block triangular pre- conditioners can be competitive to and even more efficient than the PMHSS preconditioner when they are used to accelerate Krylov subspeme iteration methods for solving block two-by-two linear systems with coefficient matrices possibly of nonsymmetric sub-blocks.展开更多
Based on the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method for the complex symmetrie linear system, two improved iterative methods, namely, the modified PMHSS (MPMHSS) method ...Based on the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method for the complex symmetrie linear system, two improved iterative methods, namely, the modified PMHSS (MPMHSS) method and the double modified PMHSS (DMPMHSS) method, are proposed in this paper. The spectra] radii of the iteration matrices of two methods are given. We show that by choosing an appropriate parameter, MPMHSS could speed up the convergence on PMHSS. The DMPMHSS method is a four-step alternating iteration that is developed upon the two-step alternating iteration of MPMHSS. We discuss the choice of the parameters and establish the convergence of DMPMHSS. In particular, we give an analysis of the spectral radius of PMHSS and DMPMHSS at the parameter free situation, and we show that DMPMHSS converges faster than PMHSS in most cases. Our numerical experiments show these points.展开更多
基金supported by the National Natural Science Foundation of China(11371275,201601D011004)
文摘在科学计算及工程应用中经常遇到复对称线性系统问题,近年来对一种特殊类型的复对称线性系统的研究已成为一个热点.基于白中治等的PMHSS方法(Bai Z Z,Benzi M,Chen F,Wang Z Q.Preconditioned MHSS iteration methods for a class of block twoby-two linear systems with applications to distributed control problems.IMA J Numer Anal,2013,33:343-369),提出一类新的PMHSS迭代法用于求解这种特殊形式的复对称线性系统,给出新方法的收敛性理论以及最优参数的表达式,最后用数值例子展示了新方法的有效性.
文摘The modified Hermitian and skew-Hermitian splitting （MHSS） iteration method and preconditioned MHSS （PMHSS） iteration method were introduced respectively. In the paper, on the basis of the MHSS iteration method, we present a PMHSS iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and complex symmetric positive definite/semi-definite matrices. Under suitable conditions, we prove the convergence of the PMHSS iteration method and discuss the spectral properties of the preconditioned matrix. Moreover, to reduce the computing cost, we establish an inexact variant of the PMHSS iteration method and analyze its convergence property in detail. Numerical results show that the PMHSS iteration method and its inexact variant are efficient and robust solvers for this class of continuous Sylvester equations.
基金supported by National Natural Science Foundation of China(Grant Nos.11021101 and 91118001)the Hundred Talent Project of Chinese Academy of Sciences and the National Basic Research Program(Grant No.2011CB309703)
文摘Based on the PMHSS preconditioning matrix, we construct a class of rotated block triangular preconditioners for block two-by-two matrices of real square blocks, and analyze the eigen-properties of the corresponding preconditioned matrices. Numerical experiments show that these rotated block triangular pre- conditioners can be competitive to and even more efficient than the PMHSS preconditioner when they are used to accelerate Krylov subspeme iteration methods for solving block two-by-two linear systems with coefficient matrices possibly of nonsymmetric sub-blocks.
文摘Based on the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method for the complex symmetrie linear system, two improved iterative methods, namely, the modified PMHSS (MPMHSS) method and the double modified PMHSS (DMPMHSS) method, are proposed in this paper. The spectra] radii of the iteration matrices of two methods are given. We show that by choosing an appropriate parameter, MPMHSS could speed up the convergence on PMHSS. The DMPMHSS method is a four-step alternating iteration that is developed upon the two-step alternating iteration of MPMHSS. We discuss the choice of the parameters and establish the convergence of DMPMHSS. In particular, we give an analysis of the spectral radius of PMHSS and DMPMHSS at the parameter free situation, and we show that DMPMHSS converges faster than PMHSS in most cases. Our numerical experiments show these points.
