In this paper,a two-step semi-regularized Hermitian and skew-Hermitian splitting(SHSS)iteration method is constructed by introducing a regularization matrix in the(1,1)-block of the first iteration step,to solve the s...In this paper,a two-step semi-regularized Hermitian and skew-Hermitian splitting(SHSS)iteration method is constructed by introducing a regularization matrix in the(1,1)-block of the first iteration step,to solve the saddle-point linear system.By carefully selecting two different regularization matrices,two kinds of SHSS preconditioners are proposed to accelerate the convergence rates of the Krylov subspace iteration methods.Theoretical analysis about the eigenvalue distribution demonstrates that the proposed SHSS preconditioners can make the eigenvalues of the corresponding preconditioned matrices be clustered around 1 and uniformly bounded away from 0.The eigenvector distribution and the upper bound on the degree of the minimal polynomial of the SHSS-preconditioned matrices indicate that the SHSS-preconditioned Krylov subspace iterative methods can converge to the true solution within finite steps in exact arithmetic.In addition,the numerical example derived from the optimal control problem shows that the SHSS preconditioners can significantly improve the convergence speeds of the Krylov subspace iteration methods,and their convergence rates are independent of the discrete mesh size.展开更多
In this paper, a generalized preconditioned Hermitian and skew-Hermitian splitting （GPHSS） iteration method for a non-Hermitian positive-definite matrix is studied, which covers standard Hermitian and skew-Hermitian...In this paper, a generalized preconditioned Hermitian and skew-Hermitian splitting （GPHSS） iteration method for a non-Hermitian positive-definite matrix is studied, which covers standard Hermitian and skew-Hermitian splitting （HSS） iteration and also many existing variants. Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix. From practical point of view, we have analyzed and implemented inexact generalized preconditioned Hermitian and skew-Hermitian splitting （IGPHSS） iteration, which employs Krylov subspace methods as its inner processes. Numerical experiments from three-dimensional convection-diffusion iterations are efficient and competitive with equation show that the GPHSS and IGPHSS standard HSS iteration and AHSS iteration.展开更多
For large and sparse saddle point problems, Zhu studied a class of generalized local Hermitian and skew-Hermitian splitting iteration methods for non-Hermitian saddle point problem [M.-Z. Zhu, Appl. Math. Comput. 218 ...For large and sparse saddle point problems, Zhu studied a class of generalized local Hermitian and skew-Hermitian splitting iteration methods for non-Hermitian saddle point problem [M.-Z. Zhu, Appl. Math. Comput. 218 （2012） 8816-8824 ]. In this paper, we further investigate the generalized local Hermitian and skew-Hermitian splitting （GLHSS） iteration methods for solving non-Hermitian generalized saddle point problems. With different choices of the parameter matrices, we derive conditions for guaranteeing the con- vergence of these iterative methods. Numerical experiments are presented to illustrate the effectiveness of our GLHSS iteration methods as well as the preconditioners.展开更多
We generalize the accelerated Hermitian and skew-Hermitian splitting(AHSS)iteration methods for large sparse saddle-point problems.These methods involve four iteration parameters whose special choices can recover the ...We generalize the accelerated Hermitian and skew-Hermitian splitting(AHSS)iteration methods for large sparse saddle-point problems.These methods involve four iteration parameters whose special choices can recover the precondi-tioned HSS and accelerated HSS iteration methods.Also a new efficient case is in-troduced and we theoretically prove that this new method converges to the unique solution of the saddle-point problem.Numerical experiments are used to further examine the effectiveness and robustness of iterations.展开更多
提出了求解广义Lyapunov方程的HSS(Hermitian and skew-Hermitian splitting)迭代法,分析了该方法的收敛性,给出了收敛因子的上界.为了降低HSS迭代法的计算量,提出了求解广义Lyapunov方程的非精确HSS迭代法,并分析其收敛性.数值结果表明...提出了求解广义Lyapunov方程的HSS(Hermitian and skew-Hermitian splitting)迭代法,分析了该方法的收敛性,给出了收敛因子的上界.为了降低HSS迭代法的计算量,提出了求解广义Lyapunov方程的非精确HSS迭代法,并分析其收敛性.数值结果表明,求解广义Lyapunov方程的HSS迭代法及非精确HSS迭代法是有效的.展开更多
对大型稀疏的非Hermite正定线性代数方程组,运用正规和反Hermite分裂(normal and skew-Hermitian splitting,NSS)迭代技巧,提出了一种两参数预处理NSS迭代法,它实际上是预处理NSS方法的推广.理论分析表明,新方法收敛于线性方程组的唯一...对大型稀疏的非Hermite正定线性代数方程组,运用正规和反Hermite分裂(normal and skew-Hermitian splitting,NSS)迭代技巧,提出了一种两参数预处理NSS迭代法,它实际上是预处理NSS方法的推广.理论分析表明,新方法收敛于线性方程组的唯一解.进一步地,推导了出现于新方法中的两个参数的最优选取,计算了对应的迭代谱的上界的最小值.新方法的实际实施中,还将不完全LU分解和增量未知元选做了两类预处理子.数值结果对所给方法的收敛性理论和有效性予以了证实.展开更多
基金the National Natural Science Foundation of China(No.12001048)R&D Program of Beijing Municipal Education Commission(No.KM202011232019),China.
文摘In this paper,a two-step semi-regularized Hermitian and skew-Hermitian splitting(SHSS)iteration method is constructed by introducing a regularization matrix in the(1,1)-block of the first iteration step,to solve the saddle-point linear system.By carefully selecting two different regularization matrices,two kinds of SHSS preconditioners are proposed to accelerate the convergence rates of the Krylov subspace iteration methods.Theoretical analysis about the eigenvalue distribution demonstrates that the proposed SHSS preconditioners can make the eigenvalues of the corresponding preconditioned matrices be clustered around 1 and uniformly bounded away from 0.The eigenvector distribution and the upper bound on the degree of the minimal polynomial of the SHSS-preconditioned matrices indicate that the SHSS-preconditioned Krylov subspace iterative methods can converge to the true solution within finite steps in exact arithmetic.In addition,the numerical example derived from the optimal control problem shows that the SHSS preconditioners can significantly improve the convergence speeds of the Krylov subspace iteration methods,and their convergence rates are independent of the discrete mesh size.
文摘In this paper, a generalized preconditioned Hermitian and skew-Hermitian splitting （GPHSS） iteration method for a non-Hermitian positive-definite matrix is studied, which covers standard Hermitian and skew-Hermitian splitting （HSS） iteration and also many existing variants. Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix. From practical point of view, we have analyzed and implemented inexact generalized preconditioned Hermitian and skew-Hermitian splitting （IGPHSS） iteration, which employs Krylov subspace methods as its inner processes. Numerical experiments from three-dimensional convection-diffusion iterations are efficient and competitive with equation show that the GPHSS and IGPHSS standard HSS iteration and AHSS iteration.
基金We would like to express our sincere gratitude to the anonymous referees whose constructive comments have the presentation of this paper greatly improved. The work was supported by the National Natural Science Foundation （No.11171371 and No.11101195）.
文摘For large and sparse saddle point problems, Zhu studied a class of generalized local Hermitian and skew-Hermitian splitting iteration methods for non-Hermitian saddle point problem [M.-Z. Zhu, Appl. Math. Comput. 218 （2012） 8816-8824 ]. In this paper, we further investigate the generalized local Hermitian and skew-Hermitian splitting （GLHSS） iteration methods for solving non-Hermitian generalized saddle point problems. With different choices of the parameter matrices, we derive conditions for guaranteeing the con- vergence of these iterative methods. Numerical experiments are presented to illustrate the effectiveness of our GLHSS iteration methods as well as the preconditioners.
文摘We generalize the accelerated Hermitian and skew-Hermitian splitting(AHSS)iteration methods for large sparse saddle-point problems.These methods involve four iteration parameters whose special choices can recover the precondi-tioned HSS and accelerated HSS iteration methods.Also a new efficient case is in-troduced and we theoretically prove that this new method converges to the unique solution of the saddle-point problem.Numerical experiments are used to further examine the effectiveness and robustness of iterations.
文摘提出了求解广义Lyapunov方程的HSS(Hermitian and skew-Hermitian splitting)迭代法,分析了该方法的收敛性,给出了收敛因子的上界.为了降低HSS迭代法的计算量,提出了求解广义Lyapunov方程的非精确HSS迭代法,并分析其收敛性.数值结果表明,求解广义Lyapunov方程的HSS迭代法及非精确HSS迭代法是有效的.
基金Project supported by the National Basic Research Program of China(973 Program,2011CB706903)the Natural Science Foundation of Jilin Province of China(201115222)
文摘对大型稀疏的非Hermite正定线性代数方程组,运用正规和反Hermite分裂(normal and skew-Hermitian splitting,NSS)迭代技巧,提出了一种两参数预处理NSS迭代法,它实际上是预处理NSS方法的推广.理论分析表明,新方法收敛于线性方程组的唯一解.进一步地,推导了出现于新方法中的两个参数的最优选取,计算了对应的迭代谱的上界的最小值.新方法的实际实施中,还将不完全LU分解和增量未知元选做了两类预处理子.数值结果对所给方法的收敛性理论和有效性予以了证实.
