Due to the difficulty in obtaining the a priori estimate,it is very hard to establish the optimal point-wise error bound of a finite difference scheme for solving a nonlinear partial differential equation in high dime...Due to the difficulty in obtaining the a priori estimate,it is very hard to establish the optimal point-wise error bound of a finite difference scheme for solving a nonlinear partial differential equation in high dimensions(2D or 3D).We here propose and analyze finite difference methods for solving the coupled GrossPitaevskii equations in two dimensions,which models the two-component Bose-Einstein condensates with an internal atomic Josephson junction.The methods which we considered include two conservative type schemes and two non-conservative type schemes.Discrete conservation laws and solvability of the schemes are analyzed.For the four proposed finite difference methods,we establish the optimal convergence rates for the error at the order of O(h^2+τ~2)in the l~∞-norm(i.e.,the point-wise error estimates)with the time stepτand the mesh size h.Besides the standard techniques of the energy method,the key techniques in the analysis is to use the cut-off function technique,transformation between the time and space direction and the method of order reduction.All the methods and results here are also valid and can be easily extended to the three-dimensional case.Finally,numerical results are reported to confirm our theoretical error estimates for the numerical methods.展开更多
In this study, the relationship between the limit of predictability and initial error was investigated using two simple chaotic systems： the Lorenz model, which possesses a single characteristic time scale, and the c...In this study, the relationship between the limit of predictability and initial error was investigated using two simple chaotic systems： the Lorenz model, which possesses a single characteristic time scale, and the coupled Lorenz model, which possesses two different characteristic time scales. The limit of predictability is defined here as the time at which the error reaches 95% of its saturation level; nonlinear behaviors of the error growth are therefore involved in the definition of the limit of predictability. Our results show that the logarithmic function performs well in describing the relationship between the limit of predictability and initial error in both models, although the coefficients in the logarithmic function were not constant across the examined range of initial errors. Compared with the Lorenz model, in the coupled Lorenz model in which the slow dynamics and the fast dynamics interact with each other--there is a more complex relationship between the limit of predictability and initial error. The limit of predictability of the Lorenz model is unbounded as the initial error becomes infinitesimally small; therefore, the limit of predictability of the Lorenz model may be extended by reducing the amplitude of the initial error. In contrast, if there exists a fixed initial error in the fast dynamics of the coupled Lorenz model, the slow dynamics has an intrinsic finite limit of predictability that cannot be extended by reducing the amplitude of the initial error in the slow dynamics, and vice versa. The findings reported here reveal the possible existence of an intrinsic finite limit of predictability in a coupled system that possesses many scales of time or motion.展开更多
In this paper, based on the study of [1], the discretizations of the coupling of finite elemellt and boundary integral are presented to solve the initial boundary value problem of parabolic partial differential equati...In this paper, based on the study of [1], the discretizations of the coupling of finite elemellt and boundary integral are presented to solve the initial boundary value problem of parabolic partial differential equation defined on an unbounded domain.The semi-discrete scheme and fully discrete scheme are given, and stability theorem and error estimates, which correspond to discrete scheme respectively,are obtained.Finally,the numerical example is provided,and numerical result shows that the method is feasible and effective.展开更多
The cross-coupled control(CCC)is widely applied to reduce contour errors in contour-following applications.In such situation,the contour error estimation plays an important role.Traditionally,the linear or second-orde...The cross-coupled control(CCC)is widely applied to reduce contour errors in contour-following applications.In such situation,the contour error estimation plays an important role.Traditionally,the linear or second-order estimation approach is adopted for biaxial motion systems,whereas only linear approach is available for triaxial systems.In this paper,the second-order contour error estimation,which was presented in our previous work,is utilized to determine the variable CCC gains for motion control systems with three axes.An integrated stable motion control strategy,which combines the feedforward,feedback and CCC controllers,is developed for multiaxis CNC systems.Experimental results on a triaxial platform indicate that the CCC scheme based on the second-order estimation,compared with that based on the linear one,significantly reduces the contour error even in the conditions of high tracking feedrate and small radius of curvature.展开更多
Optimal control of greenhouse climate is one of the key techniques in digital agriculture.Greenhouse climate,a nonlinear and uncertain system,consists of several major environmental factors such as temperature,humidit...Optimal control of greenhouse climate is one of the key techniques in digital agriculture.Greenhouse climate,a nonlinear and uncertain system,consists of several major environmental factors such as temperature,humidity,light intensity,and CO 2 concentration.Due to the complex coupled correlations,it is a challenge to achieve coordination control of greenhouse environmental factors.This paper proposes a model-free coordination control approach for greenhouse environmental factors based on Q-learning.Coordination control policy is found through systematic interaction with the dynamic environment to achieve optimal control for greenhouse climate with the control cost constraints.In order to decrease systematic trial-and-error risk and reduce the computational complexity in Q-learning algorithm,case-based reasoning （CBR） is seamlessly incorporated into the Q-learning process.The experimental results demonstrate that this approach is practical,highly effective and efficient.展开更多
Boussinesq type equations have been widely studied to model the surface water wave.In this paper,we consider the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models s...Boussinesq type equations have been widely studied to model the surface water wave.In this paper,we consider the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models such as the classical Boussinesq system,the BBM-BBM system,the Bona-Smith system,etc.We propose local discontinuous Galerkin(LDG)methods,with carefully chosen numerical fluxes,to numerically solve this abcd Boussinesq system.The main focus of this paper is to rigorously establish a priori error estimate of the proposed LDG methods for a wide range of the parameters a,b,c,d.Numerical experiments are shown to test the convergence rates,and to demonstrate that the proposed methods can simulate the head-on collision of traveling wave and finite time blow-up behavior well.展开更多
Reconciliation is a necessary step in postprocessing of continuous-variable quantum key distribution(CV-QKD)system.We use globally coupled low-density parity-check(GC-LDPC)codes in reconciliation to extract a precise ...Reconciliation is a necessary step in postprocessing of continuous-variable quantum key distribution(CV-QKD)system.We use globally coupled low-density parity-check(GC-LDPC)codes in reconciliation to extract a precise secret key from the raw keys over the authenticated classical public channel between two users.GC-LDPC codes have excellent performance over both the additive Gaussian white noise and binary-erasure channels.The reconciliation based on GC-LDPC codes can improve the reconciliation efficiency to 95.42% and reduce the frame error rate to 3.25×10^-3.Using distillation,the decoding speed can achieve 23.8 Mbits/s and decrease the cost of memory.Given decoding speed and low memory usage,this makes the proposed reconciliation method viable approach for high-speed CV-QKD system.展开更多
In this paper,a nonlinear time-fractional coupled diffusion system is solved by using a mixed finite element(MFE)method in space combined with L1-approximation and implicit second-order backward difference scheme in t...In this paper,a nonlinear time-fractional coupled diffusion system is solved by using a mixed finite element(MFE)method in space combined with L1-approximation and implicit second-order backward difference scheme in time.The stability for nonlinear fully discrete finite element scheme is analyzed and a priori error estimates are derived.Finally,some numerical tests are shown to verify our theoretical analysis.展开更多
基金supported by National Natural Science Foundation of China(Grant No.11201239)the Singapore A*STAR SERC PSF(Grant No.1321202067)
文摘Due to the difficulty in obtaining the a priori estimate,it is very hard to establish the optimal point-wise error bound of a finite difference scheme for solving a nonlinear partial differential equation in high dimensions(2D or 3D).We here propose and analyze finite difference methods for solving the coupled GrossPitaevskii equations in two dimensions,which models the two-component Bose-Einstein condensates with an internal atomic Josephson junction.The methods which we considered include two conservative type schemes and two non-conservative type schemes.Discrete conservation laws and solvability of the schemes are analyzed.For the four proposed finite difference methods,we establish the optimal convergence rates for the error at the order of O(h^2+τ~2)in the l~∞-norm(i.e.,the point-wise error estimates)with the time stepτand the mesh size h.Besides the standard techniques of the energy method,the key techniques in the analysis is to use the cut-off function technique,transformation between the time and space direction and the method of order reduction.All the methods and results here are also valid and can be easily extended to the three-dimensional case.Finally,numerical results are reported to confirm our theoretical error estimates for the numerical methods.
基金sprovided jointly by the 973 Program (Grant No.2010CB950400)National Natural Science Foundation of China (Grant Nos. 40805022 and 40821092)
文摘In this study, the relationship between the limit of predictability and initial error was investigated using two simple chaotic systems： the Lorenz model, which possesses a single characteristic time scale, and the coupled Lorenz model, which possesses two different characteristic time scales. The limit of predictability is defined here as the time at which the error reaches 95% of its saturation level; nonlinear behaviors of the error growth are therefore involved in the definition of the limit of predictability. Our results show that the logarithmic function performs well in describing the relationship between the limit of predictability and initial error in both models, although the coefficients in the logarithmic function were not constant across the examined range of initial errors. Compared with the Lorenz model, in the coupled Lorenz model in which the slow dynamics and the fast dynamics interact with each other--there is a more complex relationship between the limit of predictability and initial error. The limit of predictability of the Lorenz model is unbounded as the initial error becomes infinitesimally small; therefore, the limit of predictability of the Lorenz model may be extended by reducing the amplitude of the initial error. In contrast, if there exists a fixed initial error in the fast dynamics of the coupled Lorenz model, the slow dynamics has an intrinsic finite limit of predictability that cannot be extended by reducing the amplitude of the initial error in the slow dynamics, and vice versa. The findings reported here reveal the possible existence of an intrinsic finite limit of predictability in a coupled system that possesses many scales of time or motion.
文摘In this paper, based on the study of [1], the discretizations of the coupling of finite elemellt and boundary integral are presented to solve the initial boundary value problem of parabolic partial differential equation defined on an unbounded domain.The semi-discrete scheme and fully discrete scheme are given, and stability theorem and error estimates, which correspond to discrete scheme respectively,are obtained.Finally,the numerical example is provided,and numerical result shows that the method is feasible and effective.
基金supported by the National Natural Science Foundation of China(Grant Nos.51325502 and 51405175)the National Basic Research Program of China("973"Project)(Grant No.2011CB706804)the National Science and Technology Major Projects of China(Grant No.2012ZX04001-012-01-05)
文摘The cross-coupled control(CCC)is widely applied to reduce contour errors in contour-following applications.In such situation,the contour error estimation plays an important role.Traditionally,the linear or second-order estimation approach is adopted for biaxial motion systems,whereas only linear approach is available for triaxial systems.In this paper,the second-order contour error estimation,which was presented in our previous work,is utilized to determine the variable CCC gains for motion control systems with three axes.An integrated stable motion control strategy,which combines the feedforward,feedback and CCC controllers,is developed for multiaxis CNC systems.Experimental results on a triaxial platform indicate that the CCC scheme based on the second-order estimation,compared with that based on the linear one,significantly reduces the contour error even in the conditions of high tracking feedrate and small radius of curvature.
基金supported by National Natural Science Foundationof China(No.60775014)
文摘Optimal control of greenhouse climate is one of the key techniques in digital agriculture.Greenhouse climate,a nonlinear and uncertain system,consists of several major environmental factors such as temperature,humidity,light intensity,and CO 2 concentration.Due to the complex coupled correlations,it is a challenge to achieve coordination control of greenhouse environmental factors.This paper proposes a model-free coordination control approach for greenhouse environmental factors based on Q-learning.Coordination control policy is found through systematic interaction with the dynamic environment to achieve optimal control for greenhouse climate with the control cost constraints.In order to decrease systematic trial-and-error risk and reduce the computational complexity in Q-learning algorithm,case-based reasoning （CBR） is seamlessly incorporated into the Q-learning process.The experimental results demonstrate that this approach is practical,highly effective and efficient.
基金The work of J.Sun and Y.Xing is partially sponsored by NSF grant DMS-1753581.
文摘Boussinesq type equations have been widely studied to model the surface water wave.In this paper,we consider the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models such as the classical Boussinesq system,the BBM-BBM system,the Bona-Smith system,etc.We propose local discontinuous Galerkin(LDG)methods,with carefully chosen numerical fluxes,to numerically solve this abcd Boussinesq system.The main focus of this paper is to rigorously establish a priori error estimate of the proposed LDG methods for a wide range of the parameters a,b,c,d.Numerical experiments are shown to test the convergence rates,and to demonstrate that the proposed methods can simulate the head-on collision of traveling wave and finite time blow-up behavior well.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61801522,61972418,and 61872390)the Natural Science Foundation of Hunan Province,China(Grant Nos.2019JJ40352 and 2017JJ3415)the Special Foundation for Distinguished Young Scientists of Changsha City,China(Grant No.kq1905058)。
文摘Reconciliation is a necessary step in postprocessing of continuous-variable quantum key distribution(CV-QKD)system.We use globally coupled low-density parity-check(GC-LDPC)codes in reconciliation to extract a precise secret key from the raw keys over the authenticated classical public channel between two users.GC-LDPC codes have excellent performance over both the additive Gaussian white noise and binary-erasure channels.The reconciliation based on GC-LDPC codes can improve the reconciliation efficiency to 95.42% and reduce the frame error rate to 3.25×10^-3.Using distillation,the decoding speed can achieve 23.8 Mbits/s and decrease the cost of memory.Given decoding speed and low memory usage,this makes the proposed reconciliation method viable approach for high-speed CV-QKD system.
基金the National Natural Science Fund(11661058,11301258,11361035)the Natural Science Fund of Inner Mongolia Autonomous Region(2016MS0102,2015MS0101)+1 种基金the Scientific Research Projection of Higher Schools of Inner Mongolia(NJZZ12011)the National Undergraduate Innovative Training Project(201510126026).
文摘In this paper,a nonlinear time-fractional coupled diffusion system is solved by using a mixed finite element(MFE)method in space combined with L1-approximation and implicit second-order backward difference scheme in time.The stability for nonlinear fully discrete finite element scheme is analyzed and a priori error estimates are derived.Finally,some numerical tests are shown to verify our theoretical analysis.
