The restarted FOM method presented by Simoncini[7]according to the natural collinearity of all residuals is an efficient method for solving shifted systems,which generates the same Krylov subspace when the shifts are ...The restarted FOM method presented by Simoncini[7]according to the natural collinearity of all residuals is an efficient method for solving shifted systems,which generates the same Krylov subspace when the shifts are handled simultaneously.However,restarting slows down the convergence.We present a practical method for solving the shifted systems by adding some Ritz vectors into the Krylov subspace to form an augmented Krylov subspace. Numerical experiments illustrate that the augmented FOM approach(restarted version)can converge more quickly than the restarted FOM method.展开更多
In this paper, a complex parameter is employed in the Hermitian and skew-Hermitian splitting （HSS） method （Bai, Golub and Ng： SIAM J. Matrix Anal. Appl., 24（2003）, 603-626） for solving the complex linear system...In this paper, a complex parameter is employed in the Hermitian and skew-Hermitian splitting （HSS） method （Bai, Golub and Ng： SIAM J. Matrix Anal. Appl., 24（2003）, 603-626） for solving the complex linear system Ax = f. The convergence of the resulting method is proved when the spectrum of the matrix A lie in the right upper （or lower） part of the complex plane. We also derive an upper bound of the spectral radius of the HSS iteration matrix, and a estimated optimal parameter a （denoted by a^st） of this upper bound is presented. Numerical experiments on two modified model problems show that the HSS method with a est has a smaller spectral radius than that with the real parameter which minimizes the corresponding upper hound. In particular, for the ＇dominant＇ imaginary part of the matrix A, this improvement is considerable. We also test the GMRES method preconditioned by the HSS preconditioning matrix with our parameter a est.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
In this paper,we present the backward stochastic Taylor expansions for a Ito process,including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions.We construct the general full implicit strong T...In this paper,we present the backward stochastic Taylor expansions for a Ito process,including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions.We construct the general full implicit strong Taylor approximations(including Ito-Taylor and Stratonovich-Taylor schemes)with implicitness in both the deterministic and the stochastic terms for the stiff stochastic differential equations(SSDE)by employing truncations of backward stochastic Taylor expansions.We demonstrate that these schemes will converge strongly with corresponding order 1,2,3,....Mean-square stability has been investigated for full implicit strong Stratonovich-Taylor scheme with order 2,and it has larger meansquare stability region than the explicit and the semi-implicit strong Stratonovich-Taylor schemes with order 2.We can improve the stability of simulations considerably without too much additional computational effort by using our full implicit schemes.The full implicit strong Taylor schemes allow a larger range of time step sizes than other schemes and are suitable for SSDE with stiffness on both the drift and the diffusion terms.Our numerical experiment show these points.展开更多
We discuss the convergence property of the Lanczos method for solving a complex shifted Hermitian linear system （αI ＋ H）x = f. By showing the colinear coefficient of two system＇s residuals, our convergence analys...We discuss the convergence property of the Lanczos method for solving a complex shifted Hermitian linear system （αI ＋ H）x = f. By showing the colinear coefficient of two system＇s residuals, our convergence analysis reveals that under the condition Re（α） ＋ λmin（H） 〉 0, the method converges faster than that for the real shifted Hermitian linear system （Re（α）I ＋ H）x = f. Numerical experiments verify such convergence property.展开更多
基金Supported-by the National Natural Science Foundation of China(19971057)the Science and Technology Developing Foundation of University in Shanghai of China(02AK41).
文摘The restarted FOM method presented by Simoncini[7]according to the natural collinearity of all residuals is an efficient method for solving shifted systems,which generates the same Krylov subspace when the shifts are handled simultaneously.However,restarting slows down the convergence.We present a practical method for solving the shifted systems by adding some Ritz vectors into the Krylov subspace to form an augmented Krylov subspace. Numerical experiments illustrate that the augmented FOM approach(restarted version)can converge more quickly than the restarted FOM method.
文摘In this paper, a complex parameter is employed in the Hermitian and skew-Hermitian splitting （HSS） method （Bai, Golub and Ng： SIAM J. Matrix Anal. Appl., 24（2003）, 603-626） for solving the complex linear system Ax = f. The convergence of the resulting method is proved when the spectrum of the matrix A lie in the right upper （or lower） part of the complex plane. We also derive an upper bound of the spectral radius of the HSS iteration matrix, and a estimated optimal parameter a （denoted by a^st） of this upper bound is presented. Numerical experiments on two modified model problems show that the HSS method with a est has a smaller spectral radius than that with the real parameter which minimizes the corresponding upper hound. In particular, for the ＇dominant＇ imaginary part of the matrix A, this improvement is considerable. We also test the GMRES method preconditioned by the HSS preconditioning matrix with our parameter a est.
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.
基金supported by the Fundamental Research Funds for the Central Universities of China,and the second author is supported by the National Natural Fund Projects of China(Nos.11771100,12071332).
文摘In this paper,we present the backward stochastic Taylor expansions for a Ito process,including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions.We construct the general full implicit strong Taylor approximations(including Ito-Taylor and Stratonovich-Taylor schemes)with implicitness in both the deterministic and the stochastic terms for the stiff stochastic differential equations(SSDE)by employing truncations of backward stochastic Taylor expansions.We demonstrate that these schemes will converge strongly with corresponding order 1,2,3,....Mean-square stability has been investigated for full implicit strong Stratonovich-Taylor scheme with order 2,and it has larger meansquare stability region than the explicit and the semi-implicit strong Stratonovich-Taylor schemes with order 2.We can improve the stability of simulations considerably without too much additional computational effort by using our full implicit schemes.The full implicit strong Taylor schemes allow a larger range of time step sizes than other schemes and are suitable for SSDE with stiffness on both the drift and the diffusion terms.Our numerical experiment show these points.
文摘We discuss the convergence property of the Lanczos method for solving a complex shifted Hermitian linear system （αI ＋ H）x = f. By showing the colinear coefficient of two system＇s residuals, our convergence analysis reveals that under the condition Re（α） ＋ λmin（H） 〉 0, the method converges faster than that for the real shifted Hermitian linear system （Re（α）I ＋ H）x = f. Numerical experiments verify such convergence property.
