In this paper, we provide and analyze a new scaled conjugate gradient method and its performance, based on the modified secant equation of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method and on a new modified nonmo...In this paper, we provide and analyze a new scaled conjugate gradient method and its performance, based on the modified secant equation of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method and on a new modified nonmonotone line search technique. The method incorporates the modified BFGS secant equation in an effort to include the second order information of the objective function. The new secant equation has both gradient and function value information, and its update formula inherits the positive definiteness of Hessian approximation for general convex function. In order to improve the likelihood of finding a global optimal solution, we introduce a new modified nonmonotone line search technique. It is shown that, for nonsmooth convex problems, the proposed algorithm is globally convergent. Numerical results show that this new scaled conjugate gradient algorithm is promising and efficient for solving not only convex but also some large scale nonsmooth nonconvex problems in the sense of the Dolan-Moré performance profiles.展开更多
In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explai...In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explained by its solution in frequency domain.Furthermore,the problem is formulated into a minimization problem with a modified Tikhonov regularization method.The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem.The error estimates for the regularized solutions are obtained under Hp norm priori bound assumptions.Finally,numerical examples illustrate the effectiveness of the proposed method.展开更多
The non-linearity problem of aircraft system could not be overcome by using the MEMS sensor only.In order to improve the accuracy of aerial vehicle attitude,an aircraft attitude estimation of the MEMS sensor based on ...The non-linearity problem of aircraft system could not be overcome by using the MEMS sensor only.In order to improve the accuracy of aerial vehicle attitude,an aircraft attitude estimation of the MEMS sensor based on modified particle filter is proposed.The aircraft attitude is optimized by the conjugate gradient method,and the drift error of gyroscope is reduced.Moreover,the particle weight is updated by the observed value to obtain an optimized state estimate.Finally,the conjugate gradient method and the modified particle filter are weightily combined to determine the optimal weighting factor.The attitude estimation is carried out with STM32 and MEMS sensor as the core to design system.The experimental results show that the static and dynamic attitude estimation performances of the aircraft are improved.The performances are well,the attitude data is relatively stable,and the tracking characteristics are better.Moreover,it has better robustness and stability.展开更多
A modified polynomial preserving gradient recovery technique is proposed. Unlike the polynomial preserving gradient recovery technique,the gradient recovered with the modified polynomial preserving recovery(MPPR) is c...A modified polynomial preserving gradient recovery technique is proposed. Unlike the polynomial preserving gradient recovery technique,the gradient recovered with the modified polynomial preserving recovery(MPPR) is constructed element-wise, and it is discontinuous across the interior edges.One advantage of the MPPR technique is that the implementation is easier when adaptive meshes are involved.Superconvergence results of the gradient recovered with MPPR are proved for finite element methods for elliptic boundary problems and eigenvalue problems under adaptive meshes. The MPPR is applied to adaptive finite element methods to construct asymptotic exact a posteriori error estimates.Numerical tests are provided to examine the theoretical results and the effectiveness of the adaptive finite element algorithms.展开更多
To increase the variety and security of communication, we present the definitions of modified projective synchronization with complex scaling factors （CMPS） of real chaotic systems and complex chaotic systems, where...To increase the variety and security of communication, we present the definitions of modified projective synchronization with complex scaling factors （CMPS） of real chaotic systems and complex chaotic systems, where complex scaling factors establish a link between real chaos and complex chaos. Considering all situations of unknown parameters and pseudo-gradient condition, we design adaptive CMPS schemes based on the speed-gradient method for the real drive chaotic system and complex response chaotic system and for the complex drive chaotic system and the real response chaotic system, respectively. The convergence factors and dynamical control strength are added to regulate the convergence speed and increase robustness. Numerical simulations verify the feasibility and effectiveness of the presented schemes.展开更多
In this paper,we propose a modified proximal gradient method for solving a class of nonsmooth convex optimization problems,which arise in many contemporary statistical and signal processing applications.The proposed m...In this paper,we propose a modified proximal gradient method for solving a class of nonsmooth convex optimization problems,which arise in many contemporary statistical and signal processing applications.The proposed method adopts a new scheme to construct the descent direction based on the proximal gradient method.It is proven that the modified proximal gradient method is Q-linearly convergent without the assumption of the strong convexity of the objective function.Some numerical experiments have been conducted to evaluate the proposed method eventually.展开更多
In this paper,a robust modified weak Galerkin(MWG)finite element method for reaction-diffusion equations is proposed and investigated.An advantage of this method is that it can deal with the singularly perturbed react...In this paper,a robust modified weak Galerkin(MWG)finite element method for reaction-diffusion equations is proposed and investigated.An advantage of this method is that it can deal with the singularly perturbed reaction-diffusion equations.Another advantage of this method is that it produces fewer degrees of freedom than the traditional WG method by eliminating the element boundaries freedom.It is worth pointing out that,in our method,the test functions space is the same as the finite element space,which is helpful for the error analysis.Optimalorder error estimates are established for the corresponding numerical approximation in various norms.Some numerical results are reported to confirm the theory.展开更多
文摘In this paper, we provide and analyze a new scaled conjugate gradient method and its performance, based on the modified secant equation of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method and on a new modified nonmonotone line search technique. The method incorporates the modified BFGS secant equation in an effort to include the second order information of the objective function. The new secant equation has both gradient and function value information, and its update formula inherits the positive definiteness of Hessian approximation for general convex function. In order to improve the likelihood of finding a global optimal solution, we introduce a new modified nonmonotone line search technique. It is shown that, for nonsmooth convex problems, the proposed algorithm is globally convergent. Numerical results show that this new scaled conjugate gradient algorithm is promising and efficient for solving not only convex but also some large scale nonsmooth nonconvex problems in the sense of the Dolan-Moré performance profiles.
基金Supported by the National Natural Science Foundation of China(Grant No.11471253 and No.11571311)
文摘In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explained by its solution in frequency domain.Furthermore,the problem is formulated into a minimization problem with a modified Tikhonov regularization method.The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem.The error estimates for the regularized solutions are obtained under Hp norm priori bound assumptions.Finally,numerical examples illustrate the effectiveness of the proposed method.
基金National Natural Science Foundation of China(No.61261029)
文摘The non-linearity problem of aircraft system could not be overcome by using the MEMS sensor only.In order to improve the accuracy of aerial vehicle attitude,an aircraft attitude estimation of the MEMS sensor based on modified particle filter is proposed.The aircraft attitude is optimized by the conjugate gradient method,and the drift error of gyroscope is reduced.Moreover,the particle weight is updated by the observed value to obtain an optimized state estimate.Finally,the conjugate gradient method and the modified particle filter are weightily combined to determine the optimal weighting factor.The attitude estimation is carried out with STM32 and MEMS sensor as the core to design system.The experimental results show that the static and dynamic attitude estimation performances of the aircraft are improved.The performances are well,the attitude data is relatively stable,and the tracking characteristics are better.Moreover,it has better robustness and stability.
基金supported by the national basic research program of China under grant 2005CB321701the program for the new century outstanding talents in universities of China.
文摘A modified polynomial preserving gradient recovery technique is proposed. Unlike the polynomial preserving gradient recovery technique,the gradient recovered with the modified polynomial preserving recovery(MPPR) is constructed element-wise, and it is discontinuous across the interior edges.One advantage of the MPPR technique is that the implementation is easier when adaptive meshes are involved.Superconvergence results of the gradient recovered with MPPR are proved for finite element methods for elliptic boundary problems and eigenvalue problems under adaptive meshes. The MPPR is applied to adaptive finite element methods to construct asymptotic exact a posteriori error estimates.Numerical tests are provided to examine the theoretical results and the effectiveness of the adaptive finite element algorithms.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61273088,10971120,and 61001099)the Natural Science Foundation of Shandong Province,China(Grant No.ZR2010FM010)
文摘To increase the variety and security of communication, we present the definitions of modified projective synchronization with complex scaling factors （CMPS） of real chaotic systems and complex chaotic systems, where complex scaling factors establish a link between real chaos and complex chaos. Considering all situations of unknown parameters and pseudo-gradient condition, we design adaptive CMPS schemes based on the speed-gradient method for the real drive chaotic system and complex response chaotic system and for the complex drive chaotic system and the real response chaotic system, respectively. The convergence factors and dynamical control strength are added to regulate the convergence speed and increase robustness. Numerical simulations verify the feasibility and effectiveness of the presented schemes.
基金the National Natural Science Foundation of China(No.61179033).
文摘In this paper,we propose a modified proximal gradient method for solving a class of nonsmooth convex optimization problems,which arise in many contemporary statistical and signal processing applications.The proposed method adopts a new scheme to construct the descent direction based on the proximal gradient method.It is proven that the modified proximal gradient method is Q-linearly convergent without the assumption of the strong convexity of the objective function.Some numerical experiments have been conducted to evaluate the proposed method eventually.
基金supported by the State Key Program of National Natural Science Foundation of China(Grant 11931003)the National Natural Science Foundation of China(Grants 41974133,11971410)the Natural Science Foundation of Lingnan Normal University(Grant ZL2038).
文摘In this paper,a robust modified weak Galerkin(MWG)finite element method for reaction-diffusion equations is proposed and investigated.An advantage of this method is that it can deal with the singularly perturbed reaction-diffusion equations.Another advantage of this method is that it produces fewer degrees of freedom than the traditional WG method by eliminating the element boundaries freedom.It is worth pointing out that,in our method,the test functions space is the same as the finite element space,which is helpful for the error analysis.Optimalorder error estimates are established for the corresponding numerical approximation in various norms.Some numerical results are reported to confirm the theory.
