We propose a novel numerical approach for delay differential equations with vanishing proportional delays based on spectral methods. A Legendre-collocation method is employed to obtain highly accurate numerical approx...We propose a novel numerical approach for delay differential equations with vanishing proportional delays based on spectral methods. A Legendre-collocation method is employed to obtain highly accurate numerical approximations to the exact solution. It is proved theoretically and demonstrated numerically that the proposed method converges exponentially provided that the data in the are smooth. given pantograph delay differential equation展开更多
This work is concerned with spectrM Jacobi-collocation methods for Volterra integral equations of the second kind with a weakly singular of the form （t - s）-a When the underlying solutions are sufficiently smooth, t...This work is concerned with spectrM Jacobi-collocation methods for Volterra integral equations of the second kind with a weakly singular of the form （t - s）-a When the underlying solutions are sufficiently smooth, the convergence analysis was carried out in [Chen ＆ Tang, J. Comput. Appl. Math., 233 （2009）, pp. 938-950]; due to technical reasons 1 In this work, we will improve the results to the the results are restricted to 0 〈 μ 〈 1/2. general case 0 〈 μ 〈 1 and demonstrate that the numericl errors decay exponentially in the infinity and weighted norms when the smooth solution is involved.展开更多
This paper discusses the order-preserving convergence for spectral approximation of the self-adjoint completely continuous operator T.Under the condition that the approximate operator Th converges to T in norm,it is p...This paper discusses the order-preserving convergence for spectral approximation of the self-adjoint completely continuous operator T.Under the condition that the approximate operator Th converges to T in norm,it is proven that the k-th eigenvalue of Th converges to the k-th eigenvalue of T.(We sorted the positive eigenvalues in decreasing order and negative eigenvalues in increasing order.) Then we apply this result to conforming elements,nonconforming elements and mixed elements of self-adjoint elliptic differential operators eigenvalue problems,and prove that the k-th approximate eigenvalue obtained by these methods converges to the k-th exact eigenvalue.展开更多
This work is to provide general spectral and pseudo-spectral Jacobi-Petrov-Galerkin approaches for the second kind Volterra integro-differential equations.The Gauss-Legendre quadrature formula is used to approximate t...This work is to provide general spectral and pseudo-spectral Jacobi-Petrov-Galerkin approaches for the second kind Volterra integro-differential equations.The Gauss-Legendre quadrature formula is used to approximate the integral operator and the inner product based on the Jacobi weight is implemented in the weak formulation in the numerical implementation.For some spectral and pseudo-spectral Jacobi-Petrov-Galerkin methods,a rigorous error analysis in both L2_(ω^(α,β))^(2),and L^(∞)norms is given provided that both the kernel function and the source function are sufficiently smooth.Numerical experiments validate the theoretical prediction.展开更多
This paper presents a detailed review of both theory and algorithms for the Cheeger cut based on the graph 1-Laplacian. In virtue of the cell structure of the feasible set, we propose a cell descend （CD） framework f...This paper presents a detailed review of both theory and algorithms for the Cheeger cut based on the graph 1-Laplacian. In virtue of the cell structure of the feasible set, we propose a cell descend （CD） framework for achieving the Cheeger cut. While plugging the relaxation to guarantee the decrease of the objective value in the feasible set, from which both the inverse power （IP） method and the steepest descent （SD） method can also be recovered, we are able to get two specified CD methods. Comparisons of all these methods are conducted on several typical graphs.展开更多
Explicit convergence rates in geometric and strong ergodicity for denumerable discrete time Markov chains with general reversible transition matrices are obtained in terms of the geometric moments or uniform moments o...Explicit convergence rates in geometric and strong ergodicity for denumerable discrete time Markov chains with general reversible transition matrices are obtained in terms of the geometric moments or uniform moments of the hitting times to a fixed point.Another way by Lyapunov's drift conditions is also used to derive these convergence rates.As a typical example,the discrete time birth-death process(random walk) is studied and the explicit criteria for geometric ergodicity are presented.展开更多
We describe the application of the spectral method to delay integro-differential equations with proportional delays. It is shown that the resulting numerical solutions exhibit the spectral convergence order. Extension...We describe the application of the spectral method to delay integro-differential equations with proportional delays. It is shown that the resulting numerical solutions exhibit the spectral convergence order. Extensions to equations with more general （nonlinear） vanishing delays are also discussed.展开更多
Abstract Let P be a transition matrix which is symmetric with respect to a measure π. The spectral gap of P in L2（π）-space, denoted by gap（P）, is defined as the distance between 1 and the rest of the spectrum of...Abstract Let P be a transition matrix which is symmetric with respect to a measure π. The spectral gap of P in L2（π）-space, denoted by gap（P）, is defined as the distance between 1 and the rest of the spectrum of P. In this paper, we study the relationship between gap（P） and the convergence rate of P^n. When P is transient, the convergence rate of pn is equal to 1 - gap（P）. When P is ergodic, we give the explicit upper and lower bounds for the convergence rate of pn in terms of gap（P）. These results are extended to L^∞ （π）-space.展开更多
We propose and analyze a single-interval Legendre-Gauss-Radau(LGR)spectral collocation method for nonlinear second-order initial value problems of ordinary differential equations.We design an efficient iterative algor...We propose and analyze a single-interval Legendre-Gauss-Radau(LGR)spectral collocation method for nonlinear second-order initial value problems of ordinary differential equations.We design an efficient iterative algorithm and prove spectral convergence for the single-interval LGR collocation method.For more effective implementation,we propose a multi-interval LGR spectral collocation scheme,which provides us great flexibility with respect to the local time steps and local approximation degrees.Moreover,we combine the multi-interval LGR collocation method in time with the Legendre-Gauss-Lobatto collocation method in space to obtain a space-time spectral collocation approximation for nonlinear second-order evolution equations.Numerical results show that the proposed methods have high accuracy and excellent long-time stability.Numerical comparison between our methods and several commonly used methods are also provided.展开更多
The theory of a class of spectral methods is extended to Volterra integrodifferential equations which contain a weakly singular kernel(t−s)^(−μ) with 0<μ<1.In this work,we consider the case when the underlying...The theory of a class of spectral methods is extended to Volterra integrodifferential equations which contain a weakly singular kernel(t−s)^(−μ) with 0<μ<1.In this work,we consider the case when the underlying solutions of weakly singular Volterra integro-differential equations are sufficiently smooth.We provide a rigorous error analysis for the spectral methods,which shows that both the errors of approximate solutions and the errors of approximate derivatives of the solutions decay exponentially in L^(∞)-norm and weighted L^(2)-norm.The numerical examples are given to illustrate the theoretical results.展开更多
In this work,we analyze the three-step backward differentiation formula(BDF3)method for solving the Allen-Cahn equation on variable grids.For BDF2 method,the discrete orthogonal convolution(DOC)kernels are positive,th...In this work,we analyze the three-step backward differentiation formula(BDF3)method for solving the Allen-Cahn equation on variable grids.For BDF2 method,the discrete orthogonal convolution(DOC)kernels are positive,the stability and convergence analysis are well established in[Liao and Zhang,Math.Comp.,90(2021),1207–1226]and[Chen,Yu,and Zhang,arXiv:2108.02910,2021].However,the numerical analysis for BDF3 method with variable steps seems to be highly nontrivial due to the additional degrees of freedom and the non-positivity of DOC kernels.By developing a novel spectral norm inequality,the unconditional stability and convergence are rigorously proved under the updated step ratio restriction rk:=τk/τk−1≤1.405 for BDF3 method.Finally,numerical experiments are performed to illustrate the theoretical results.To the best of our knowledge,this is the first theoretical analysis of variable steps BDF3 method for the Allen-Cahn equation.展开更多
As a generalization of the two-term conjugate gradient method(CGM),the spectral CGM is one of the effective methods for solving unconstrained optimization.In this paper,we enhance the JJSL conjugate parameter,initiall...As a generalization of the two-term conjugate gradient method(CGM),the spectral CGM is one of the effective methods for solving unconstrained optimization.In this paper,we enhance the JJSL conjugate parameter,initially proposed by Jiang et al.(Computational and Applied Mathematics,2021,40:174),through the utilization of a convex combination technique.And this improvement allows for an adaptive search direction by integrating a newly constructed spectral gradient-type restart strategy.Then,we develop a new spectral CGM by employing an inexact line search to determine the step size.With the application of the weak Wolfe line search,we establish the sufficient descent property of the proposed search direction.Moreover,under general assumptions,including the employment of the strong Wolfe line search for step size calculation,we demonstrate the global convergence of our new algorithm.Finally,the given unconstrained optimization test results show that the new algorithm is effective.展开更多
基金The research of HB was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada and by the Research Grants Council of Hong KongThe research of TT was supported by Hong Kong Baptist University,the Research Grants Council of Hong Kong and he was supported in part by the Chinese Academy of Sciences while visiting its Institute of Computational Mathematics.
文摘We propose a novel numerical approach for delay differential equations with vanishing proportional delays based on spectral methods. A Legendre-collocation method is employed to obtain highly accurate numerical approximations to the exact solution. It is proved theoretically and demonstrated numerically that the proposed method converges exponentially provided that the data in the are smooth. given pantograph delay differential equation
基金Acknowledgments. This work is supported by National Science Foundation of China （1127114 5）, Foundation for Talent Introduction of Guangdong Provincial University, Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme （2008）, Specialized Research Fund for the Doctoral Program of Higher Education （20114407110009）, and the Project of Department of Education of Guangdong Province （No. [2012] 290）. The second author is sup- ported by the Natural Science Foundation of Fujian Province, China （2012J01007） and Start-up fund of Fuzhou University （0460022456）. The second and third author are supported by the FRG Grant of Hong Kong Baptist University and the RGC Grants provided by Research Grant Council of Hong Kong.
文摘This work is concerned with spectrM Jacobi-collocation methods for Volterra integral equations of the second kind with a weakly singular of the form （t - s）-a When the underlying solutions are sufficiently smooth, the convergence analysis was carried out in [Chen ＆ Tang, J. Comput. Appl. Math., 233 （2009）, pp. 938-950]; due to technical reasons 1 In this work, we will improve the results to the the results are restricted to 0 〈 μ 〈 1/2. general case 0 〈 μ 〈 1 and demonstrate that the numericl errors decay exponentially in the infinity and weighted norms when the smooth solution is involved.
基金supported by the National Natural Science Foundation of China (Grant No. 10761003)Guizhou Province Scientific Research for Senior Personnels
文摘This paper discusses the order-preserving convergence for spectral approximation of the self-adjoint completely continuous operator T.Under the condition that the approximate operator Th converges to T in norm,it is proven that the k-th eigenvalue of Th converges to the k-th eigenvalue of T.(We sorted the positive eigenvalues in decreasing order and negative eigenvalues in increasing order.) Then we apply this result to conforming elements,nonconforming elements and mixed elements of self-adjoint elliptic differential operators eigenvalue problems,and prove that the k-th approximate eigenvalue obtained by these methods converges to the k-th exact eigenvalue.
基金supported by the National Natural Science Foundation of China(10871066)Project of Scientific Research Fund of Hunan Provincial Education Department(09K025)+2 种基金Programme for New Century Excellent Talents in University(NCET-06-0712)supported by the Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Provincesupported in part by Natural Science Foundation of Guizhou Province(LKS[2010]05).
文摘This work is to provide general spectral and pseudo-spectral Jacobi-Petrov-Galerkin approaches for the second kind Volterra integro-differential equations.The Gauss-Legendre quadrature formula is used to approximate the integral operator and the inner product based on the Jacobi weight is implemented in the weak formulation in the numerical implementation.For some spectral and pseudo-spectral Jacobi-Petrov-Galerkin methods,a rigorous error analysis in both L2_(ω^(α,β))^(2),and L^(∞)norms is given provided that both the kernel function and the source function are sufficiently smooth.Numerical experiments validate the theoretical prediction.
文摘This paper presents a detailed review of both theory and algorithms for the Cheeger cut based on the graph 1-Laplacian. In virtue of the cell structure of the feasible set, we propose a cell descend （CD） framework for achieving the Cheeger cut. While plugging the relaxation to guarantee the decrease of the objective value in the feasible set, from which both the inverse power （IP） method and the steepest descent （SD） method can also be recovered, we are able to get two specified CD methods. Comparisons of all these methods are conducted on several typical graphs.
基金supported by Program for New Century Excellent Talents in University,National Basic Research Program of China (973 Project) (Grant No.2006CB805901)National Natural Science Foundation of China (Grant No.10721091)
文摘Explicit convergence rates in geometric and strong ergodicity for denumerable discrete time Markov chains with general reversible transition matrices are obtained in terms of the geometric moments or uniform moments of the hitting times to a fixed point.Another way by Lyapunov's drift conditions is also used to derive these convergence rates.As a typical example,the discrete time birth-death process(random walk) is studied and the explicit criteria for geometric ergodicity are presented.
文摘We describe the application of the spectral method to delay integro-differential equations with proportional delays. It is shown that the resulting numerical solutions exhibit the spectral convergence order. Extensions to equations with more general （nonlinear） vanishing delays are also discussed.
基金Supported in part by 985 Project,973 Project(Grant No.2011CB808000)National Natural Science Foundation of China(Grant No.11131003)+1 种基金Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20100003110005)the Fundamental Research Funds for the Central Universities
文摘Abstract Let P be a transition matrix which is symmetric with respect to a measure π. The spectral gap of P in L2（π）-space, denoted by gap（P）, is defined as the distance between 1 and the rest of the spectrum of P. In this paper, we study the relationship between gap（P） and the convergence rate of P^n. When P is transient, the convergence rate of pn is equal to 1 - gap（P）. When P is ergodic, we give the explicit upper and lower bounds for the convergence rate of pn in terms of gap（P）. These results are extended to L^∞ （π）-space.
基金supported in part by the National Natural Science Foundation of China(Grant Nos.12171322,11771298 and 11871043)the Natural Science Foundation of Shanghai(Grant Nos.21ZR1447200,20ZR1441200 and 22ZR1445500)the Science and Technology Innovation Plan of Shanghai(Grant No.20JC1414200).
文摘We propose and analyze a single-interval Legendre-Gauss-Radau(LGR)spectral collocation method for nonlinear second-order initial value problems of ordinary differential equations.We design an efficient iterative algorithm and prove spectral convergence for the single-interval LGR collocation method.For more effective implementation,we propose a multi-interval LGR spectral collocation scheme,which provides us great flexibility with respect to the local time steps and local approximation degrees.Moreover,we combine the multi-interval LGR collocation method in time with the Legendre-Gauss-Lobatto collocation method in space to obtain a space-time spectral collocation approximation for nonlinear second-order evolution equations.Numerical results show that the proposed methods have high accuracy and excellent long-time stability.Numerical comparison between our methods and several commonly used methods are also provided.
基金This work is supported by the Foundation for Talent Introduction of Guangdong Provincial University,Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2008)National Science Foundation of China(10971074).
文摘The theory of a class of spectral methods is extended to Volterra integrodifferential equations which contain a weakly singular kernel(t−s)^(−μ) with 0<μ<1.In this work,we consider the case when the underlying solutions of weakly singular Volterra integro-differential equations are sufficiently smooth.We provide a rigorous error analysis for the spectral methods,which shows that both the errors of approximate solutions and the errors of approximate derivatives of the solutions decay exponentially in L^(∞)-norm and weighted L^(2)-norm.The numerical examples are given to illustrate the theoretical results.
基金supported by the Science Fund for Distinguished Young Scholars of Gansu Province(Grant No.23JRRA1020)the Fundamental Research Funds for the Central Universities(Grant No.lzujbky-2023-06).
文摘In this work,we analyze the three-step backward differentiation formula(BDF3)method for solving the Allen-Cahn equation on variable grids.For BDF2 method,the discrete orthogonal convolution(DOC)kernels are positive,the stability and convergence analysis are well established in[Liao and Zhang,Math.Comp.,90(2021),1207–1226]and[Chen,Yu,and Zhang,arXiv:2108.02910,2021].However,the numerical analysis for BDF3 method with variable steps seems to be highly nontrivial due to the additional degrees of freedom and the non-positivity of DOC kernels.By developing a novel spectral norm inequality,the unconditional stability and convergence are rigorously proved under the updated step ratio restriction rk:=τk/τk−1≤1.405 for BDF3 method.Finally,numerical experiments are performed to illustrate the theoretical results.To the best of our knowledge,this is the first theoretical analysis of variable steps BDF3 method for the Allen-Cahn equation.
基金supported by the National Natural Science Foundation of China(No.72071202)the Key Laboratory of Mathematics and Engineering Applications,Ministry of Education。
文摘As a generalization of the two-term conjugate gradient method(CGM),the spectral CGM is one of the effective methods for solving unconstrained optimization.In this paper,we enhance the JJSL conjugate parameter,initially proposed by Jiang et al.(Computational and Applied Mathematics,2021,40:174),through the utilization of a convex combination technique.And this improvement allows for an adaptive search direction by integrating a newly constructed spectral gradient-type restart strategy.Then,we develop a new spectral CGM by employing an inexact line search to determine the step size.With the application of the weak Wolfe line search,we establish the sufficient descent property of the proposed search direction.Moreover,under general assumptions,including the employment of the strong Wolfe line search for step size calculation,we demonstrate the global convergence of our new algorithm.Finally,the given unconstrained optimization test results show that the new algorithm is effective.
