A new direct method for solving unsymmetrical sparse linear systems(USLS) arising from meshless methods was introduced. Computation of certain meshless methods such as meshless local Petrov-Galerkin (MLPG) method ...A new direct method for solving unsymmetrical sparse linear systems(USLS) arising from meshless methods was introduced. Computation of certain meshless methods such as meshless local Petrov-Galerkin (MLPG) method need to solve large USLS. The proposed solution method for unsymmetrical case performs factorization processes symmetrically on the upper and lower triangular portion of matrix, which differs from previous work based on general unsymmetrical process, and attains higher performance. It is shown that the solution algorithm for USLS can be simply derived from the existing approaches for the symmetrical case. The new matrix factorization algorithm in our method can be implemented easily by modifying a standard JKI symmetrical matrix factorization code. Multi-blocked out-of-core strategies were also developed to expand the solution scale. The approach convincingly increases the speed of the solution process, which is demonstrated with the numerical tests.展开更多
Fast solving large-scale linear equations in the finite element analysis is a classical subject in computational mechanics. It is a key technique in computer aided engineering (CAE) and computer aided manufacturing ...Fast solving large-scale linear equations in the finite element analysis is a classical subject in computational mechanics. It is a key technique in computer aided engineering (CAE) and computer aided manufacturing (CAM). This paper presents a high-efficiency improved symmetric successive over-relaxation (ISSOR) preconditioned conjugate gradient (PCG) method, which maintains lelism consistent with the original form. Ideally, the by 50% as compared with the original algorithm. the convergence and inherent paralcomputation can It is suitable for be reduced nearly high-performance computing with its inherent basic high-efficiency operations. By comparing with the numerical results, it is shown that the proposed method has the best performance.展开更多
In this paper,we study the hyperstability for the general linear equation f(ax+by)=Af(x)+Bf(y)in the setting of complete quasi-2-Banach spaces.We first extend the main fixed point result of Brzdek and Ciepliński(Acta...In this paper,we study the hyperstability for the general linear equation f(ax+by)=Af(x)+Bf(y)in the setting of complete quasi-2-Banach spaces.We first extend the main fixed point result of Brzdek and Ciepliński(Acta Mathematica Scientia,2018,38 B(2):377-390)to quasi-2-Banach spaces by defining an equivalent quasi-2-Banach space.Then we use this result to generalize the main results on the hyperstability for the general linear equation in quasi-2-Banach spaces.Our results improve and generalize many results of literature.展开更多
Let K be a proper cone in R^x,let A be an n×n real matrix that satisfies AK(?)K,letb be a given vector of K,and let λbe a given positive real number.The following two lin-ear equations are considered in this pap...Let K be a proper cone in R^x,let A be an n×n real matrix that satisfies AK(?)K,letb be a given vector of K,and let λbe a given positive real number.The following two lin-ear equations are considered in this paper:(i)(λⅠ_n-A)x=b,x∈K,and(ii)(A-λⅠ_n)x=b,x∈K.We obtain several equivalent conditions for the solvability of the first equation.展开更多
In this paper,we consider the high order method for solving the linear transport equations under diffusive scaling and with random inputs.To tackle the randomness in the problem,the stochastic Galerkin method of the g...In this paper,we consider the high order method for solving the linear transport equations under diffusive scaling and with random inputs.To tackle the randomness in the problem,the stochastic Galerkin method of the generalized polynomial chaos approach has been employed.Besides,the high order implicit-explicit scheme under the micro-macro decomposition framework and the discontinuous Galerkin method have been employed.We provide several numerical experiments to validate the accuracy and the stochastic asymptotic-preserving property.展开更多
This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either...This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.展开更多
In this work,we study the linearized Landau equation with soft potentials and show that the smooth solution to the Cauchy problem with initial datum in L^(2)(ℝ^(3))enjoys an analytic regularization effect,and that the...In this work,we study the linearized Landau equation with soft potentials and show that the smooth solution to the Cauchy problem with initial datum in L^(2)(ℝ^(3))enjoys an analytic regularization effect,and that the evolution of the analytic radius is the same as the heat equations.展开更多
Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of ...Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.展开更多
Based on a new efficient identification technique of active constraints introduced in this paper, a new sequential systems of linear equations (SSLE) algorithm generating feasible iterates is proposed for solving no...Based on a new efficient identification technique of active constraints introduced in this paper, a new sequential systems of linear equations (SSLE) algorithm generating feasible iterates is proposed for solving nonlinear optimization problems with inequality constraints. In this paper, we introduce a new technique for constructing the system of linear equations, which recurs to a perturbation for the gradients of the constraint functions. At each iteration of the new algorithm, a feasible descent direction is obtained by solving only one system of linear equations without doing convex combination. To ensure the global convergence and avoid the Maratos effect, the algorithm needs to solve two additional reduced systems of linear equations with the same coefficient matrix after finite iterations. The proposed algorithm is proved to be globally and superlinearly convergent under some mild conditions. What distinguishes this algorithm from the previous feasible SSLE algorithms is that an improving direction is obtained easily and the computation cost of generating a new iterate is reduced. Finally, a preliminary implementation has been tested.展开更多
Two kinds of iterative methods are designed to solve the linear system of equations, we obtain a new interpretation in terms of a geometric concept. Therefore, we have a better insight into the essence of the iterativ...Two kinds of iterative methods are designed to solve the linear system of equations, we obtain a new interpretation in terms of a geometric concept. Therefore, we have a better insight into the essence of the iterative methods and provide a reference for further study and design. Finally, a new iterative method is designed named as the diverse relaxation parameter of the SOR method which, in particular, demonstrates the geometric characteristics. Many examples prove that the method is quite effective.展开更多
Finding the solution to a general multivariate modular linear equation plays an important role in cryptanalysis field. Earlier results show that obtaining a relatively short solution is possible in polynomial time. Ho...Finding the solution to a general multivariate modular linear equation plays an important role in cryptanalysis field. Earlier results show that obtaining a relatively short solution is possible in polynomial time. However, one problem arises here that if the equation has a short solution in given bounded range, the results outputted by earlier algorithms are often not the ones we are interested in. In this paper, we present a probability method based on lattice basis reduction to solve the problem. For a general multivariate modular linear equation with short solution in the given bounded range, the new method outputs this short solution in polynomial time, with a high probability. When the number of unknowns is not too large (smaller than 68), the probability is approximating 1. Experimental results show that Knapsack systems and Lu-Lee type systems are easily broken in polynomial time with this new method.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
In this paper,we study the multivariate linear equations with arbitrary positive integral coefficients.Under the Generalized Riemann Hypothesis,we obtained the asymptotic formula for the linear equations with more tha...In this paper,we study the multivariate linear equations with arbitrary positive integral coefficients.Under the Generalized Riemann Hypothesis,we obtained the asymptotic formula for the linear equations with more than five prime variables.This asymptotic formula is composed of three parts,that is,the first main term,the explicit second main term and the error term.Among them,the first main term is similar with the former one,the explicit second main term is relative to the non-trivial zeros of Dirichlet L-functions,and our error term improves the former one.展开更多
In this paper,we consider the simultaneous representation of pairs of integers as linear combinations in three prime variables and obtain a related numerical bound.
Presents information on a study which proposed a superlinearly convergent algorithm of sequential systems of linear equations or nonlinear optimization problems with inequality constraints. Assumptions; Discussion on ...Presents information on a study which proposed a superlinearly convergent algorithm of sequential systems of linear equations or nonlinear optimization problems with inequality constraints. Assumptions; Discussion on lemmas about several matrices related to the common coefficient matrix F; Strengthening of the regularity assumptions on the functions involved; Numerical experiments.展开更多
In this paper, we consider solving dense linear equations on Dawning1000 byusing matrix partitioning technique. Based on this partitioning of matrix, we give aparallel block LU decomposition method. The efficiency of ...In this paper, we consider solving dense linear equations on Dawning1000 byusing matrix partitioning technique. Based on this partitioning of matrix, we give aparallel block LU decomposition method. The efficiency of solving linear equationsby different ways is analysed. The numerical results are given on Dawning1000.By running our parallel program, the best speed up on 32 processors is over 25.展开更多
In this article, we study the complex oscillation problems of entire solutions to homogeneous and nonhomogeneous linear difference equations, and obtain some relations of the exponent of convergence of zeros and the o...In this article, we study the complex oscillation problems of entire solutions to homogeneous and nonhomogeneous linear difference equations, and obtain some relations of the exponent of convergence of zeros and the order of growth of entire solutions to complex linear difference equations.展开更多
Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are ...Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.展开更多
In this article, the authors study the growth of certain second order linear differential equation f″+A(z)f′+B(z)f=0 and give precise estimates for the hyperorder of solutions of infinite order. Under similar ...In this article, the authors study the growth of certain second order linear differential equation f″+A(z)f′+B(z)f=0 and give precise estimates for the hyperorder of solutions of infinite order. Under similar conditions, higher order differential equations will be considered.展开更多
基金Project supported by the National Natural Science Foundation of China (Nos. 10232040, 10572002 and 10572003)
文摘A new direct method for solving unsymmetrical sparse linear systems(USLS) arising from meshless methods was introduced. Computation of certain meshless methods such as meshless local Petrov-Galerkin (MLPG) method need to solve large USLS. The proposed solution method for unsymmetrical case performs factorization processes symmetrically on the upper and lower triangular portion of matrix, which differs from previous work based on general unsymmetrical process, and attains higher performance. It is shown that the solution algorithm for USLS can be simply derived from the existing approaches for the symmetrical case. The new matrix factorization algorithm in our method can be implemented easily by modifying a standard JKI symmetrical matrix factorization code. Multi-blocked out-of-core strategies were also developed to expand the solution scale. The approach convincingly increases the speed of the solution process, which is demonstrated with the numerical tests.
基金Project supported by the National Natural Science Foundation of China(Nos.5130926141030747+3 种基金41102181and 51121005)the National Basic Research Program of China(973 Program)(No.2011CB013503)the Young Teachers’ Initial Funding Scheme of Sun Yat-sen University(No.39000-1188140)
文摘Fast solving large-scale linear equations in the finite element analysis is a classical subject in computational mechanics. It is a key technique in computer aided engineering (CAE) and computer aided manufacturing (CAM). This paper presents a high-efficiency improved symmetric successive over-relaxation (ISSOR) preconditioned conjugate gradient (PCG) method, which maintains lelism consistent with the original form. Ideally, the by 50% as compared with the original algorithm. the convergence and inherent paralcomputation can It is suitable for be reduced nearly high-performance computing with its inherent basic high-efficiency operations. By comparing with the numerical results, it is shown that the proposed method has the best performance.
基金AISTDF,DST India for the research grant vide project No.CRD/2018/000017。
文摘In this paper,we study the hyperstability for the general linear equation f(ax+by)=Af(x)+Bf(y)in the setting of complete quasi-2-Banach spaces.We first extend the main fixed point result of Brzdek and Ciepliński(Acta Mathematica Scientia,2018,38 B(2):377-390)to quasi-2-Banach spaces by defining an equivalent quasi-2-Banach space.Then we use this result to generalize the main results on the hyperstability for the general linear equation in quasi-2-Banach spaces.Our results improve and generalize many results of literature.
文摘Let K be a proper cone in R^x,let A be an n×n real matrix that satisfies AK(?)K,letb be a given vector of K,and let λbe a given positive real number.The following two lin-ear equations are considered in this paper:(i)(λⅠ_n-A)x=b,x∈K,and(ii)(A-λⅠ_n)x=b,x∈K.We obtain several equivalent conditions for the solvability of the first equation.
基金supported by the Simons Foundation:Collaboration Grantssupported by the AFOSR grant FA9550-18-1-0383.
文摘In this paper,we consider the high order method for solving the linear transport equations under diffusive scaling and with random inputs.To tackle the randomness in the problem,the stochastic Galerkin method of the generalized polynomial chaos approach has been employed.Besides,the high order implicit-explicit scheme under the micro-macro decomposition framework and the discontinuous Galerkin method have been employed.We provide several numerical experiments to validate the accuracy and the stochastic asymptotic-preserving property.
基金supported by the NSF under Grant DMS-2208391sponsored by the NSF under Grant DMS-1753581.
文摘This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.
基金supported by the Natural Science Foundation of Hubei Province,China (2022CFB444)the Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA)+1 种基金supported by the NSFC (12031006)the Fundamental Research Funds for the Central Universities of China.
文摘In this work,we study the linearized Landau equation with soft potentials and show that the smooth solution to the Cauchy problem with initial datum in L^(2)(ℝ^(3))enjoys an analytic regularization effect,and that the evolution of the analytic radius is the same as the heat equations.
文摘Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.
基金Supported by National Natural Science Foundation of China (Grant No. 10771040)Guangxi Science Foundation (Grant No. 0832052)Guangxi University for Nationalities Youth Foundation (Grant No. 2007QN24)
文摘Based on a new efficient identification technique of active constraints introduced in this paper, a new sequential systems of linear equations (SSLE) algorithm generating feasible iterates is proposed for solving nonlinear optimization problems with inequality constraints. In this paper, we introduce a new technique for constructing the system of linear equations, which recurs to a perturbation for the gradients of the constraint functions. At each iteration of the new algorithm, a feasible descent direction is obtained by solving only one system of linear equations without doing convex combination. To ensure the global convergence and avoid the Maratos effect, the algorithm needs to solve two additional reduced systems of linear equations with the same coefficient matrix after finite iterations. The proposed algorithm is proved to be globally and superlinearly convergent under some mild conditions. What distinguishes this algorithm from the previous feasible SSLE algorithms is that an improving direction is obtained easily and the computation cost of generating a new iterate is reduced. Finally, a preliminary implementation has been tested.
基金Supported by the National Natural Science Foundation of China(61272300)
文摘Two kinds of iterative methods are designed to solve the linear system of equations, we obtain a new interpretation in terms of a geometric concept. Therefore, we have a better insight into the essence of the iterative methods and provide a reference for further study and design. Finally, a new iterative method is designed named as the diverse relaxation parameter of the SOR method which, in particular, demonstrates the geometric characteristics. Many examples prove that the method is quite effective.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 60873249, 60973142)the National High-Tech Research & Development Program of China (Grant Nos. 2008AA10Z419, 2009AA011906)the Project Funded by Basic Research Foundation of School of Information Science and Technology of Tsinghua University
文摘Finding the solution to a general multivariate modular linear equation plays an important role in cryptanalysis field. Earlier results show that obtaining a relatively short solution is possible in polynomial time. However, one problem arises here that if the equation has a short solution in given bounded range, the results outputted by earlier algorithms are often not the ones we are interested in. In this paper, we present a probability method based on lattice basis reduction to solve the problem. For a general multivariate modular linear equation with short solution in the given bounded range, the new method outputs this short solution in polynomial time, with a high probability. When the number of unknowns is not too large (smaller than 68), the probability is approximating 1. Experimental results show that Knapsack systems and Lu-Lee type systems are easily broken in polynomial time with this new method.
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In this paper,we study the multivariate linear equations with arbitrary positive integral coefficients.Under the Generalized Riemann Hypothesis,we obtained the asymptotic formula for the linear equations with more than five prime variables.This asymptotic formula is composed of three parts,that is,the first main term,the explicit second main term and the error term.Among them,the first main term is similar with the former one,the explicit second main term is relative to the non-trivial zeros of Dirichlet L-functions,and our error term improves the former one.
基金Project supported by the National Natural Science Foundation of Chinathe Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry+2 种基金Shanghai's Shuguang Projectthe Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE,P. R. ChinaPartially supported by the Natural Sciences and Engineering Research Council of Canada
文摘In this paper,we consider the simultaneous representation of pairs of integers as linear combinations in three prime variables and obtain a related numerical bound.
基金This research was supported by the National Natural Science Foundation of China(19571001, 19971002, 79970014) Cross-century Excellent Personnel Award and Teaching and Research Award Program for Outstanding Young Teachers in High Education Ministry o
文摘Presents information on a study which proposed a superlinearly convergent algorithm of sequential systems of linear equations or nonlinear optimization problems with inequality constraints. Assumptions; Discussion on lemmas about several matrices related to the common coefficient matrix F; Strengthening of the regularity assumptions on the functions involved; Numerical experiments.
文摘In this paper, we consider solving dense linear equations on Dawning1000 byusing matrix partitioning technique. Based on this partitioning of matrix, we give aparallel block LU decomposition method. The efficiency of solving linear equationsby different ways is analysed. The numerical results are given on Dawning1000.By running our parallel program, the best speed up on 32 processors is over 25.
基金supported by the National Natural Science Foundation of China (11171119 and 10871076)
文摘In this article, we study the complex oscillation problems of entire solutions to homogeneous and nonhomogeneous linear difference equations, and obtain some relations of the exponent of convergence of zeros and the order of growth of entire solutions to complex linear difference equations.
文摘Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.
基金the National Natural Science Foundation of China(10161006,10571044)the Natural Science Foundation of Guangdong Prov(06025059)
文摘In this article, the authors study the growth of certain second order linear differential equation f″+A(z)f′+B(z)f=0 and give precise estimates for the hyperorder of solutions of infinite order. Under similar conditions, higher order differential equations will be considered.