The preconditioned methods for solving linear system are discussed. The convergence rate of accelerated overrelaxation (AOR) method can be enlarged by using the preconditioned method when the classical AOR method conv...The preconditioned methods for solving linear system are discussed. The convergence rate of accelerated overrelaxation (AOR) method can be enlarged by using the preconditioned method when the classical AOR method converges, and the preconditioned method is invalid when the classical iterative method does not converge. The results in corresponding references are improved and perfected.展开更多
A class of general inverse matrix techniques based on adaptive algorithmic modelling methodologies is derived yielding iterative methods for solving unsymmetric linear systems of irregular structure arising in complex...A class of general inverse matrix techniques based on adaptive algorithmic modelling methodologies is derived yielding iterative methods for solving unsymmetric linear systems of irregular structure arising in complex computational problems in three space dimensions. The proposed class of approximate inverse is chosen as the basis to yield systems on which classic and preconditioned iterative methods are explicitly applied. Optimized versions of the proposed approximate inverse are presented using special storage (k-sweep) techniques leading to economical forms of the approximate inverses. Application of the adaptive algorithmic methodologies on a characteristic nonlinear boundary value problem is discussed and numerical results are given.展开更多
Recently, some authors (Li, Yang and Wu, 2014) studied the parameterized preconditioned HSS (PPHSS) method for solving saddle point problems. In this short note, we further discuss the PPHSS method for solving singula...Recently, some authors (Li, Yang and Wu, 2014) studied the parameterized preconditioned HSS (PPHSS) method for solving saddle point problems. In this short note, we further discuss the PPHSS method for solving singular saddle point problems. We prove the semi-convergence of the PPHSS method under some conditions. Numerical experiments are given to illustrate the efficiency of the method with appropriate parameters.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
We establish local convergence results for a generic algorithmic framework for solving a wide class of equality constrained optimization problems.The framework is based on applying a splitting scheme to the augmented ...We establish local convergence results for a generic algorithmic framework for solving a wide class of equality constrained optimization problems.The framework is based on applying a splitting scheme to the augmented Lagrangian function that includes as a special case the well-known alternating direction method of multipliers(ADMM).Our local convergence analysis is free of the usual restrictions on ADMM-like methods,such as convexity,block separability or linearity of constraints.It offers a much-needed theoretical justification to the widespread practice of applying ADMM-like methods to nonconvex optimization problems.展开更多
We discuss estimates for the rate of convergence of the method of successive subspace corrections in terms of condition number estimate for the method of parallel subspace corrections.We provide upper bounds and in a ...We discuss estimates for the rate of convergence of the method of successive subspace corrections in terms of condition number estimate for the method of parallel subspace corrections.We provide upper bounds and in a special case,a lower bound for preconditioners defined via the method of successive subspace corrections.展开更多
文章考虑具有更优特性的分块矩阵,(具有性质A的矩阵),给出了预条件Jacob i、G auss-Se ide l、对称G auss-Se ide l迭代矩阵与传统块Jacob i迭代矩阵二者特征值之间的关系,作为应用,选取某个恰当的预条件因子,在传统块Jacob i迭代法不...文章考虑具有更优特性的分块矩阵,(具有性质A的矩阵),给出了预条件Jacob i、G auss-Se ide l、对称G auss-Se ide l迭代矩阵与传统块Jacob i迭代矩阵二者特征值之间的关系,作为应用,选取某个恰当的预条件因子,在传统块Jacob i迭代法不收敛的情况下,预条件块迭代法能收敛.展开更多
文摘The preconditioned methods for solving linear system are discussed. The convergence rate of accelerated overrelaxation (AOR) method can be enlarged by using the preconditioned method when the classical AOR method converges, and the preconditioned method is invalid when the classical iterative method does not converge. The results in corresponding references are improved and perfected.
文摘A class of general inverse matrix techniques based on adaptive algorithmic modelling methodologies is derived yielding iterative methods for solving unsymmetric linear systems of irregular structure arising in complex computational problems in three space dimensions. The proposed class of approximate inverse is chosen as the basis to yield systems on which classic and preconditioned iterative methods are explicitly applied. Optimized versions of the proposed approximate inverse are presented using special storage (k-sweep) techniques leading to economical forms of the approximate inverses. Application of the adaptive algorithmic methodologies on a characteristic nonlinear boundary value problem is discussed and numerical results are given.
文摘Recently, some authors (Li, Yang and Wu, 2014) studied the parameterized preconditioned HSS (PPHSS) method for solving saddle point problems. In this short note, we further discuss the PPHSS method for solving singular saddle point problems. We prove the semi-convergence of the PPHSS method under some conditions. Numerical experiments are given to illustrate the efficiency of the method with appropriate parameters.
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.
基金This work was supported in part by Shenzhen Fundamental Research Fund(Nos.JCYJ-20170306141038939,KQJSCX-20170728162302784,ZDSYS-201707251409055)via the Shenzhen Research Institute of Big DataThe work of Jun-Feng Yang was supported by the National Natural Science Foundation of China(Nos.11771208,11922111,11671195).
文摘We establish local convergence results for a generic algorithmic framework for solving a wide class of equality constrained optimization problems.The framework is based on applying a splitting scheme to the augmented Lagrangian function that includes as a special case the well-known alternating direction method of multipliers(ADMM).Our local convergence analysis is free of the usual restrictions on ADMM-like methods,such as convexity,block separability or linearity of constraints.It offers a much-needed theoretical justification to the widespread practice of applying ADMM-like methods to nonconvex optimization problems.
文摘We discuss estimates for the rate of convergence of the method of successive subspace corrections in terms of condition number estimate for the method of parallel subspace corrections.We provide upper bounds and in a special case,a lower bound for preconditioners defined via the method of successive subspace corrections.
文摘文章考虑具有更优特性的分块矩阵,(具有性质A的矩阵),给出了预条件Jacob i、G auss-Se ide l、对称G auss-Se ide l迭代矩阵与传统块Jacob i迭代矩阵二者特征值之间的关系,作为应用,选取某个恰当的预条件因子,在传统块Jacob i迭代法不收敛的情况下,预条件块迭代法能收敛.
