Numerical treatment of engineering application problems often eventually results in a solution of systems of linear or nonlinear equations.The solution process using digital computational devices usually takes tremend...Numerical treatment of engineering application problems often eventually results in a solution of systems of linear or nonlinear equations.The solution process using digital computational devices usually takes tremendous time due to the extremely large size encountered in most real-world engineering applications.So,practical solvers for systems of linear and nonlinear equations based on multi graphic process units（GPUs）are proposed in order to accelerate the solving process.In the linear and nonlinear solvers,the preconditioned bi-conjugate gradient stable（PBi-CGstab）method and the Inexact Newton method are used to achieve the fast and stable convergence behavior.Multi-GPUs are utilized to obtain more data storage that large size problems need.展开更多
In this paper, we provide a generalized block-by-block method for constructing block-by-block systems to solve the system of linear Volterra integral equations of the second kind, and then deduce some of the special c...In this paper, we provide a generalized block-by-block method for constructing block-by-block systems to solve the system of linear Volterra integral equations of the second kind, and then deduce some of the special cases. Compared with the expansion method and He＇s homotopy perturbation method, respectively numerical examples are given to certify the effectiveness of the method. The results show that the block-by-block method is very effective, simple, and of high accuracy in solving the system of linear Volterra integral equations of the second kind.展开更多
The convergence of the parallel matrix multisplitting relaxation methods presented by Wang (Linear Algebra and Its Applications 154/156 (1991) 473 486) is further investigated. The investigations show that these relax...The convergence of the parallel matrix multisplitting relaxation methods presented by Wang (Linear Algebra and Its Applications 154/156 (1991) 473 486) is further investigated. The investigations show that these relaxation methods really have considerably larger convergence domains.展开更多
We present a class of asymptotically optimal successive overrelaxation methods for solving the large sparse system of linear equations. Numerical computations show that these new methods are more efficient and robust ...We present a class of asymptotically optimal successive overrelaxation methods for solving the large sparse system of linear equations. Numerical computations show that these new methods are more efficient and robust than the classical successive overrelaxation method.展开更多
We establish the convergence theories of the symmetric relaxation methods for the system of linear equations with symmetric positive definite coefficient matrix, and more generally, those of the unsymmetric relaxation...We establish the convergence theories of the symmetric relaxation methods for the system of linear equations with symmetric positive definite coefficient matrix, and more generally, those of the unsymmetric relaxation methods for the system of linear equations with positive definite matrix.展开更多
Based on the greedy randomized Kaczmarz(GRK)method,we propose a multi-step greedy Kaczmarz method for solving large-scale consistent linear systems,utilizing multi-step projection techniques.Its convergence is proved ...Based on the greedy randomized Kaczmarz(GRK)method,we propose a multi-step greedy Kaczmarz method for solving large-scale consistent linear systems,utilizing multi-step projection techniques.Its convergence is proved when the linear system is consistent.Numerical experiments demonstrate that the proposed method is effective and more efficient than several existing classical Kaczmarz methods.展开更多
Presents preconditioning matrices having parallel computing function for the coefficient matrix and a class of parallel hybrid algebraic multilevel iteration methods for solving linear equations. Solution to elliptic ...Presents preconditioning matrices having parallel computing function for the coefficient matrix and a class of parallel hybrid algebraic multilevel iteration methods for solving linear equations. Solution to elliptic boundary value problem; Discussion on symmetric positive definite matrix; Computational complexities.展开更多
A class of asynchronous matrix multi-splitting multi-parameter relaxation methods, including the asynchronous matrix multisplitting SAOR, SSOR and SGS methods as well. as the known asynchronous matrix multisplitting A...A class of asynchronous matrix multi-splitting multi-parameter relaxation methods, including the asynchronous matrix multisplitting SAOR, SSOR and SGS methods as well. as the known asynchronous matrix multisplitting AOR, SOR and GS methods, etc., is proposed for solving the large sparse systems of linear equations by making use of the principle of sufficiently using the delayed information. These new methods can greatly execute the parallel computational efficiency of the MIMD-systems, and are shown to be convergent when the coefficient matrices are H-matrices. Moreover, necessary and sufficient conditions ensuring the convergence of these methods are concluded for the case that the coefficient matrices are L-matrices.展开更多
In this paper, an improved feasible QP-free method is proposed to solve nonlinear inequality constrained optimization problems. Here, a new modified method is presented to obtain the revised feasible descent direction...In this paper, an improved feasible QP-free method is proposed to solve nonlinear inequality constrained optimization problems. Here, a new modified method is presented to obtain the revised feasible descent direction. In view of the computational cost, the most attractive feature of the new algorithm is that only one system of linear equations is required to obtain the revised feasible descent direction. Thereby, per single iteration, it is only necessary to solve three systems of linear equations with the same coefficient matrix. In particular, without the positive definiteness assumption on the Hessian estimate, the proposed algorithm is still global convergence. Under some suitable conditions, the superlinear convergence rate is obtained.展开更多
In this paper,a new generalized m-CMP inverse is introduced into the Minkowski space,and its related properties are discussed.Secondly,three different expressions of m-CMP inverse are listed,namely Hartwig-Spindel-boc...In this paper,a new generalized m-CMP inverse is introduced into the Minkowski space,and its related properties are discussed.Secondly,three different expressions of m-CMP inverse are listed,namely Hartwig-Spindel-bock decomposition,full-rank decomposition and integral expression.Finally,the application of the inverse m-CMP equation in solving linear equations is given.展开更多
文摘Numerical treatment of engineering application problems often eventually results in a solution of systems of linear or nonlinear equations.The solution process using digital computational devices usually takes tremendous time due to the extremely large size encountered in most real-world engineering applications.So,practical solvers for systems of linear and nonlinear equations based on multi graphic process units（GPUs）are proposed in order to accelerate the solving process.In the linear and nonlinear solvers,the preconditioned bi-conjugate gradient stable（PBi-CGstab）method and the Inexact Newton method are used to achieve the fast and stable convergence behavior.Multi-GPUs are utilized to obtain more data storage that large size problems need.
基金Supported by the National Natural Science Foundation of China(10962008)
文摘In this paper, we provide a generalized block-by-block method for constructing block-by-block systems to solve the system of linear Volterra integral equations of the second kind, and then deduce some of the special cases. Compared with the expansion method and He＇s homotopy perturbation method, respectively numerical examples are given to certify the effectiveness of the method. The results show that the block-by-block method is very effective, simple, and of high accuracy in solving the system of linear Volterra integral equations of the second kind.
文摘The convergence of the parallel matrix multisplitting relaxation methods presented by Wang (Linear Algebra and Its Applications 154/156 (1991) 473 486) is further investigated. The investigations show that these relaxation methods really have considerably larger convergence domains.
基金Subsidized by the Special Funds For Major State Basic Research Project G1999032803.
文摘We present a class of asymptotically optimal successive overrelaxation methods for solving the large sparse system of linear equations. Numerical computations show that these new methods are more efficient and robust than the classical successive overrelaxation method.
文摘We establish the convergence theories of the symmetric relaxation methods for the system of linear equations with symmetric positive definite coefficient matrix, and more generally, those of the unsymmetric relaxation methods for the system of linear equations with positive definite matrix.
基金supported by the National Natural Science Foundation of China(No.12071149)the Science and Technology Commission of Shanghai Municipality of China(No.22DZ2229014)the National Key R&D Program of China(No.2022YFA1004403).
文摘Based on the greedy randomized Kaczmarz(GRK)method,we propose a multi-step greedy Kaczmarz method for solving large-scale consistent linear systems,utilizing multi-step projection techniques.Its convergence is proved when the linear system is consistent.Numerical experiments demonstrate that the proposed method is effective and more efficient than several existing classical Kaczmarz methods.
基金Subsidized by the Special Funds for Major State Basic Research Projects G1999032803 and Suported bythe National Natural Scienc
文摘Presents preconditioning matrices having parallel computing function for the coefficient matrix and a class of parallel hybrid algebraic multilevel iteration methods for solving linear equations. Solution to elliptic boundary value problem; Discussion on symmetric positive definite matrix; Computational complexities.
基金Project 19601036 supported by the National Natural Science Foundation of China.
文摘A class of asynchronous matrix multi-splitting multi-parameter relaxation methods, including the asynchronous matrix multisplitting SAOR, SSOR and SGS methods as well. as the known asynchronous matrix multisplitting AOR, SOR and GS methods, etc., is proposed for solving the large sparse systems of linear equations by making use of the principle of sufficiently using the delayed information. These new methods can greatly execute the parallel computational efficiency of the MIMD-systems, and are shown to be convergent when the coefficient matrices are H-matrices. Moreover, necessary and sufficient conditions ensuring the convergence of these methods are concluded for the case that the coefficient matrices are L-matrices.
基金Supported by National Natural Science Foundation of China (Grant Nos. 11061011 and 71061002)Guangxi Fund for Distinguished Young Scholars (2012GXSFFA060003)
文摘In this paper, an improved feasible QP-free method is proposed to solve nonlinear inequality constrained optimization problems. Here, a new modified method is presented to obtain the revised feasible descent direction. In view of the computational cost, the most attractive feature of the new algorithm is that only one system of linear equations is required to obtain the revised feasible descent direction. Thereby, per single iteration, it is only necessary to solve three systems of linear equations with the same coefficient matrix. In particular, without the positive definiteness assumption on the Hessian estimate, the proposed algorithm is still global convergence. Under some suitable conditions, the superlinear convergence rate is obtained.
文摘In this paper,a new generalized m-CMP inverse is introduced into the Minkowski space,and its related properties are discussed.Secondly,three different expressions of m-CMP inverse are listed,namely Hartwig-Spindel-bock decomposition,full-rank decomposition and integral expression.Finally,the application of the inverse m-CMP equation in solving linear equations is given.
