In recent years,many efforts have been made to numerically solving the constrained optimization distributed control problems,in which the most common one is to discretize the partial differential equation first and th...In recent years,many efforts have been made to numerically solving the constrained optimization distributed control problems,in which the most common one is to discretize the partial differential equation first and then solve the resulting system of linear equations.A number of preconditioned Krylov subspace methods have been constructed to solve the resulting system of linear equations in the literature.In this paper,by analyzing the block-diagonal preconditioner presented by Zhang,et al.(Zhang X Y,Yan H Y,Huang Y M.On preconditioned MINRES method for solving the distributed control problems.Commun Appl Math Comput,2014,28:128-132.),we propose a parameterized block-diagonally preconditioned linear system where a parameterized preconditioner is utilized and the preconditioned MINRES method is applied to solve the system of linear equations.The spectral analysis of the proposed preconditioned matrix shows that the spectral distribution of the parameterized preconditioning matrix should be much more clustered if the parameter is greater than 1.Numerical Experiments show that the preconditioned MINRES method is efficient for solving the distributed control problems.展开更多
In this paper, we investigate the block Lanczos algorithm for solving large sparse symmetric linear systems with multiple right-hand sides, and show how to incorporate deflation to drop converged linear systems using ...In this paper, we investigate the block Lanczos algorithm for solving large sparse symmetric linear systems with multiple right-hand sides, and show how to incorporate deflation to drop converged linear systems using a natural convergence criterion, and present an adaptive block Lanczos algorithm. We propose also a block version of Paige and Saunders’ MINRES method for iterative solution of symmetric linear systems, and describe important implementation details. We establish a relationship between the block Lanczos algorithm and block MINRES algorithm, and compare the numerical performance of the Lanczos algorithm and MINRES method for symmetric linear systems applied to a sequence of right hand sides with that of the block Lanczos algorithm and block MINRES algorithm for multiple linear systems simultaneously.[WT5,5”HZ]展开更多
The four-dimensional variational assimilation(4D-Var)has been widely used in meteorological and oceanographic data assimilation.This method is usually implemented in the model space,known as primal approach(P4D-Var).A...The four-dimensional variational assimilation(4D-Var)has been widely used in meteorological and oceanographic data assimilation.This method is usually implemented in the model space,known as primal approach(P4D-Var).Alternatively,physical space analysis system(4D-PSAS)is proposed to reduce the computation cost,in which the 4D-Var problem is solved in physical space(i.e.,observation space).In this study,the conjugate gradient(CG)algorithm,implemented in the 4D-PSAS system is evaluated and it is found that the non-monotonic change of the gradient norm of 4D-PSAS cost function causes artificial oscillations of cost function in the iteration process.The reason of non-monotonic variation of gradient norm in 4D-PSAS is then analyzed.In order to overcome the non-monotonic variation of gradient norm,a new algorithm,Minimum Residual(MINRES)algorithm,is implemented in the process of assimilation iteration in this study.Our experimental results show that the improved 4D-PSAS with the MINRES algorithm guarantees the monotonic reduction of gradient norm of cost function,greatly improves the convergence properties of 4D-PSAS as well,and significantly restrains the numerical noises associated with the traditional 4D-PSAS system.展开更多
基金Project supported by the National Natural Science Foundation of China(11571156)
文摘In recent years,many efforts have been made to numerically solving the constrained optimization distributed control problems,in which the most common one is to discretize the partial differential equation first and then solve the resulting system of linear equations.A number of preconditioned Krylov subspace methods have been constructed to solve the resulting system of linear equations in the literature.In this paper,by analyzing the block-diagonal preconditioner presented by Zhang,et al.(Zhang X Y,Yan H Y,Huang Y M.On preconditioned MINRES method for solving the distributed control problems.Commun Appl Math Comput,2014,28:128-132.),we propose a parameterized block-diagonally preconditioned linear system where a parameterized preconditioner is utilized and the preconditioned MINRES method is applied to solve the system of linear equations.The spectral analysis of the proposed preconditioned matrix shows that the spectral distribution of the parameterized preconditioning matrix should be much more clustered if the parameter is greater than 1.Numerical Experiments show that the preconditioned MINRES method is efficient for solving the distributed control problems.
文摘In this paper, we investigate the block Lanczos algorithm for solving large sparse symmetric linear systems with multiple right-hand sides, and show how to incorporate deflation to drop converged linear systems using a natural convergence criterion, and present an adaptive block Lanczos algorithm. We propose also a block version of Paige and Saunders’ MINRES method for iterative solution of symmetric linear systems, and describe important implementation details. We establish a relationship between the block Lanczos algorithm and block MINRES algorithm, and compare the numerical performance of the Lanczos algorithm and MINRES method for symmetric linear systems applied to a sequence of right hand sides with that of the block Lanczos algorithm and block MINRES algorithm for multiple linear systems simultaneously.[WT5,5”HZ]
基金The National Key Research and Development Program of China under contract Nos 2017YFC1501803 and2018YFC1506903the National Natural Science Foundation of China under contract Nos 91730304,41475021 and 41575026
文摘The four-dimensional variational assimilation(4D-Var)has been widely used in meteorological and oceanographic data assimilation.This method is usually implemented in the model space,known as primal approach(P4D-Var).Alternatively,physical space analysis system(4D-PSAS)is proposed to reduce the computation cost,in which the 4D-Var problem is solved in physical space(i.e.,observation space).In this study,the conjugate gradient(CG)algorithm,implemented in the 4D-PSAS system is evaluated and it is found that the non-monotonic change of the gradient norm of 4D-PSAS cost function causes artificial oscillations of cost function in the iteration process.The reason of non-monotonic variation of gradient norm in 4D-PSAS is then analyzed.In order to overcome the non-monotonic variation of gradient norm,a new algorithm,Minimum Residual(MINRES)algorithm,is implemented in the process of assimilation iteration in this study.Our experimental results show that the improved 4D-PSAS with the MINRES algorithm guarantees the monotonic reduction of gradient norm of cost function,greatly improves the convergence properties of 4D-PSAS as well,and significantly restrains the numerical noises associated with the traditional 4D-PSAS system.
