For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of structured preconditioners through matrix transformation and matrix approximations. For the specific versions such a...For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to high-quality preconditioning matrices for some typical matrices from the real-world applications.展开更多
An improved SSOR-like(ISSOR-like)preconditioner is proposed for the non-Hermitian positive definite linear system with a dominant skew-Hermitian part.The upper and lower bounds on the real and imaginary parts of the e...An improved SSOR-like(ISSOR-like)preconditioner is proposed for the non-Hermitian positive definite linear system with a dominant skew-Hermitian part.The upper and lower bounds on the real and imaginary parts of the eigenvalues of the ISSOR-like preconditioned matrix and the convergence property of the corresponding ISSOR-like iteration method are discussed in depth.Numerical experiments show that the ISSOR-like preconditioner can effectively accelerate preconditioned GMRES.展开更多
In this paper,we consider numerical methods for two-sided space variable-order fractional diffusion equations(VOFDEs)with a nonlinear source term.The implicit Euler(IE)method and a shifted Grünwald(SG)scheme are ...In this paper,we consider numerical methods for two-sided space variable-order fractional diffusion equations(VOFDEs)with a nonlinear source term.The implicit Euler(IE)method and a shifted Grünwald(SG)scheme are used to approximate the temporal derivative and the space variable-order(VO)fractional derivatives,respectively,which leads to an IE-SG scheme.Since the order of the VO derivatives depends on the space and the time variables,the corresponding coefficient matrices arising from the discretization of VOFDEs are dense and without the Toeplitz-like structure.In light of the off-diagonal decay property of the coefficient matrices,we consider applying the preconditioned generalized minimum residual methods with banded preconditioners to solve the discretization systems.The eigenvalue distribution and the condition number of the preconditioned matrices are studied.Numerical results show that the proposed banded preconditioners are efficient.展开更多
In this paper,we consider a modified alternating positive semidefinite splitting preconditioner for solving the saddle point problems arising from the finite element discretization of the hybrid formulation of the tim...In this paper,we consider a modified alternating positive semidefinite splitting preconditioner for solving the saddle point problems arising from the finite element discretization of the hybrid formulation of the time-harmonic eddy current model.The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are studied for both simple and general topology.Numerical results demonstrate the effectiveness of the proposed preconditioner when it is used to accelerate the convergence rate of Krylov subspace methods such as GMRES.展开更多
We outline an approach to investigate the limiting law of an absorbing Markov chain conditional on having not been absorbed for long time. The main idea is to employ Donsker-Varadhan＇s entropy functional which is typ...We outline an approach to investigate the limiting law of an absorbing Markov chain conditional on having not been absorbed for long time. The main idea is to employ Donsker-Varadhan＇s entropy functional which is typically used as the large deviation rate function for Markov processes. This approach provides an interpretation for a certain quasi-ergodicity展开更多
文摘For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to high-quality preconditioning matrices for some typical matrices from the real-world applications.
文摘An improved SSOR-like(ISSOR-like)preconditioner is proposed for the non-Hermitian positive definite linear system with a dominant skew-Hermitian part.The upper and lower bounds on the real and imaginary parts of the eigenvalues of the ISSOR-like preconditioned matrix and the convergence property of the corresponding ISSOR-like iteration method are discussed in depth.Numerical experiments show that the ISSOR-like preconditioner can effectively accelerate preconditioned GMRES.
文摘In this paper,we consider numerical methods for two-sided space variable-order fractional diffusion equations(VOFDEs)with a nonlinear source term.The implicit Euler(IE)method and a shifted Grünwald(SG)scheme are used to approximate the temporal derivative and the space variable-order(VO)fractional derivatives,respectively,which leads to an IE-SG scheme.Since the order of the VO derivatives depends on the space and the time variables,the corresponding coefficient matrices arising from the discretization of VOFDEs are dense and without the Toeplitz-like structure.In light of the off-diagonal decay property of the coefficient matrices,we consider applying the preconditioned generalized minimum residual methods with banded preconditioners to solve the discretization systems.The eigenvalue distribution and the condition number of the preconditioned matrices are studied.Numerical results show that the proposed banded preconditioners are efficient.
基金This research is supported by the National Key Research and Development Program of China(Nos.2019YFC0312003 and 2018YFC1504200)the National Natural Science Foundation of China(Nos.11901098 and U1839207).
文摘In this paper,we consider a modified alternating positive semidefinite splitting preconditioner for solving the saddle point problems arising from the finite element discretization of the hybrid formulation of the time-harmonic eddy current model.The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are studied for both simple and general topology.Numerical results demonstrate the effectiveness of the proposed preconditioner when it is used to accelerate the convergence rate of Krylov subspace methods such as GMRES.
基金Acknowledgements This work was supported by the Specialized Research Fund for the Doctoral Program of Higher Education （20120002110045） and the National Natural Science Foundation of China （Grant No. 11271220）. The author was grateful to the referees for the careful reading of the first version of the paper.
文摘We outline an approach to investigate the limiting law of an absorbing Markov chain conditional on having not been absorbed for long time. The main idea is to employ Donsker-Varadhan＇s entropy functional which is typically used as the large deviation rate function for Markov processes. This approach provides an interpretation for a certain quasi-ergodicity
