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15 October 2022, Volume 39 Issue 5 Previous Issue    Next Issue

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A Krylov Subspace Optimization Method in Artificial Neural Network ZHANG Zhenyu, LIN Muyang 2022, 39 (5):  681-694.  doi: 10.3969/j.issn.1005-3085.2022.05.001 Abstract ( 215 )   PDF (592KB) ( 126 )   Save The development of algorithms for optimizing the loss function of artificial neural networks is introduced is this work. The KSD (Krylov Subspace Descent) algorithm is extended to MKSD (Modified KSD) algorithm which has adaptively variable subspace dimension instead of fixed dimension. Some numerical examples of optimizing the fully connected neural network problems by MKSD, KSD and SGD (Stochastic Gradient Descent) algorithms are given. The numerical results show that the MKSD method has certain advantages over other methods. Related Articles | Metrics

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Objective Bayesian Analysis and Application for Zero-and-one-inflated Geometric Distribution Using Data Augmentation XIAO Xiang 2022, 39 (5):  695-708.  doi: 10.3969/j.issn.1005-3085.2022.05.002 Abstract ( 109 )   PDF (236KB) ( 337 )   Save Count data with excess zeros and ones arise frequently in many practical application fields such as traffic safety, public health, risk management and so on. In this paper, in order to study this kind of data set more deeply, a zero-and-one-inflated geometric distribution model is proposed and considered. Jeffreys prior and reference priors are derived via data augmentation strategy and the complete likelihood function. For different sample sizes and different true values of the parameters, numerical simulation and comparative analysis are adopted to assess the performance of these different objective priors. Finally, one accidental data set from Detroit is analyzed to illustrate the practicability of the proposed model. It shows that the objective Bayesian method performs better than the subjective Bayesian method. Related Articles | Metrics

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Analysis of a New Chaotic System and Its Synchronization Control SHI Shuaishuai, LIU Licai, DU Chuanhong 2022, 39 (5):  709-724.  doi: 10.3969/j.issn.1005-3085.2022.05.003 Abstract ( 101 )   PDF (7329KB) ( 42 )   Save In this paper, a new chaotic system is proposed. Through the analysis of the bifurcation diagram, Lyapunov exponent and phase diagram of the system, it is proved that the system is a chaotic system under certain parameters. When the parameters are changed, the system exhibits the alternation behavior of period--doubling period--chaos--period--doubling period--chaos, which indicates that the system has more complex dynamic characteristics and sensitivity to the initial value. In addition, the Multisim circuit simulation software is applied to verify the physical realizability of the system. Moreover, based on the new chaotic system, a mismatch projection synchronization controller is designed, which theoretically confirms that the system is synchronized with the Chen chaotic system under the action of the controller, the effectiveness of the synchronous controller is validated by Matlab simulation. Finally, the digital encrypted transmission is realized under the action of the synchronous controller. Related Articles | Metrics

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Dynamic Behavior Analysis of a Stochastic Tungiasis Epidemic Model KONG Lili, LI Luping, CHEN Huiqin, KANG Shugui 2022, 39 (5):  725-738.  doi: 10.3969/j.issn.1005-3085.2022.05.004 Abstract ( 110 )   PDF (547KB) ( 66 )   Save Tungiasis is a zoonotic disease in poverty-stricken areas, and its pathogenesis is easily affected by random fluctuation environmental factors. Therefore, a class of stochastic tungiasis model with correct hygiene habits as a control strategy is established and discussed. Firstly, the existence and uniqueness of global positive solutions of stochastic systems are proved by proper Lyapunov functions and It$\hat{\rm o}$ formula. Secondly, under certain conditions, the oscillation behavior of the positive solution of the stochastic system around the equilibrium point of the deterministic system is proved. Finally, the correctness of the theoretical analysis is verified by the numerical simulation. The results indicate that the disease will become extinct when the intensity of random interference is high enough. Related Articles | Metrics

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Capacity Expansion Problem of Spanning Arborescence with Constraints YANG Zilan, ZHU Juanping, LI Rui 2022, 39 (5):  739-749.  doi: 10.3969/j.issn.1005-3085.2022.05.005 Abstract ( 128 )   PDF (241KB) ( 64 )   Save We consider the problem of communication network expansion and upgrading, which is abstracted as the capacity expansion problem of spanning arborescence with constraints (CEPAC) in directed networks. Firstly, we prove that CEPAC problem is NP-hard by constructing an instance of CEPAC based on 0-1 knapsack problem. Secondly, by Megiddo parameter search of the spanning tree and Megiddo parameter search strategy of matroid intersection, we establish the relationship between the spanning arborescence polytope and the matroid intersection, and transform an optimal spanning arborescence into an adjacent optimal spanning arborescence through the basic transformation. Then, we design a $(2,1)$-approximate matroid intersection algorithm with constraints for CEPAC problem. Finally, we discuss the capacity expansion problem minimum arborescence (CEPMA) and solve it by modifying Chu-Liu-Edmonds algorithm with lexicographical order. Related Articles | Metrics

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Dynamic Behavior of an HIV Infection Model with Intracellular Delay XING Qinghong, LI Can, MA Huilian, GUO Zunguang 2022, 39 (5):  750-762.  doi: 10.3969/j.issn.1005-3085.2022.05.006 Abstract ( 149 )   PDF (228KB) ( 90 )   Save An HIV infection dynamics model with intracellular delay is studied. The basic reproduction ratio is obtained by calculation. By analyzing the distribution of roots of the corresponding characteristic equations, the local stability of each of feasible equilibria is established. By constructing the suitable Lyapunov functional, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable. From the persistence theory of infinite dimensional dynamic system, it shows that if the basic reproduction ratio is greater than unity, the system is uniformly persistent. According to the iteration technique and comparision arguments, the sufficient conditions are obtained for the global attractivity of the chronic infection equilibrium. Related Articles | Metrics

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Fast Sweeping WCNS for Steady State Problems for Hyperbolic Conservation Laws CHEN Xun, JIANG Yanqun, ZHANG Xu, HU Yinggang 2022, 39 (5):  763-774.  doi: 10.3969/j.issn.1005-3085.2022.05.007 Abstract ( 78 )   PDF (374KB) ( 228 )   Save Steady state problems for nonlinear hyperbolic conservation laws widely exist in many fields, such as fluid mechanics, optimal control, picture processing, and so on. A fast sweeping weighted compact nonlinear scheme (WCNS) is designed for these problems. The model equations are first transformed into the time-dependent problems by adding time derivative terms. Then, the flux derivatives are computed by a third-order WCNS and a third-order TVD Runge-Kutta method is used for the time discretization. Moreover, the fast sweeping strategy that combines Gauss-Seidel-type iteration method with a finite number of alternating sweeping orderings is adopted to accelerate the convergence speed of the designed algorithm. Numerical results show that the fast sweeping WCNS has third-order accuracy and good shock-capturing ability. Compared with the TVD Runge-Kutta WCNS without sweeping, the fast sweeping WCNS has a faster convergence speed and higher computational efficiency. Related Articles | Metrics

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The Adaptive Finite Element Method for Elliptic Optimal Control Problem in H (curl) Space HE Zhilong, ZHAO Jianping, YANG Huan, LI Bing, XI Mengru 2022, 39 (5):  775-796.  doi: 10.3969/j.issn.1005-3085.2022.05.008 Abstract ( 59 )   PDF (697KB) ( 75 )   Save In this paper, we present and analyze the adaptive finite element method for elliptic optimal control problem in H (curl) space. Firstly, based on Maxwell equation, we propose the elliptical optimal control problem in H (curl) space and establish an optimal control system. The optimal control model is equivalently transformed into the PDE systems and present the regularity property of the OCM. Then we apply the AFEM to solve the system. For the finite element approximation, we introduce the limitations of the posterior error estimator of the residual type and prove the convergence. The numerical examples are presented to verify theoretical results and indicate that the AFEM is more reliable and effective under the same conditions. Finally, our theoretical analysis and numerical algorithms can be promoted and applied to more complex problems. Related Articles | Metrics

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Cell-centered Finite Volume Scheme for Evolutionary Diffusion Equations on Arbitrary Polygonal Meshes SHAN Li, JIN Zhu, ZHANG Haicheng 2022, 39 (5):  797-812.  doi: 10.3969/j.issn.1005-3085.2022.05.009 Abstract ( 90 )   PDF (306KB) ( 70 )   Save A cell-centered finite volume scheme for the 2D evolutionary diffusion equation on arbitrary polygonal meshes is constructed. We apply the backward Euler scheme to discrete the time derivative term, and employ the vertex unknowns as auxiliary ones to discrete the diffusion operator, by solving an underdetermined linear system of equations, vertex unknowns can be expressed by a linear combination of the central unknowns, which finally results in a cell-centered scheme. The proposed scheme maintains the local conservation and the linearity preserving properties. Considering the continuous and discontinuous diffusion coefficients respectively, several numerical experiments on different kinds of polygonal meshes show that second-order convergence rate can be obtained. Its numerical performance is significantly better than the nine point scheme with arithmetic average weighting and inverse distance weighting, and is similar to the weighting method of bilinear interpolation, it overcomes the disadvantage that bilinear interpolation is not suitable for triangular meshes. Besides, the numerical results also implies that proposed scheme can still achieve second-order convergence for solving nonlinear diffusion equations. Related Articles | Metrics

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The Least Squares $\eta$-Hermitian Problems of Split Quaternion Matrix Equation $AXB+CYD=E$ ZHANG Ying, WANG Weihua, WEI Jianing, ZHANG Huisheng 2022, 39 (5):  813-825.  doi: 10.3969/j.issn.1005-3085.2022.05.010 Abstract ( 171 )   PDF (196KB) ( 180 )   Save The split quaternion constrained matrix equation is an important problem in mathematical research and physical applications. To overcome the difficulty in obtaining the least-squares solution of the split quaternion matrix equation, the least-squares $\eta$-Hermitian solution of the split quaternion matrix equation $AXB + CYD = E$ is considered. Firstly, anti-involution transformation and $\eta$-Hermitian matrix based on split quaternion are defined. Secondly, the Frobenius norm of a split quaternion matrix is introduced based on the complex representation of a split quaternion matrix, which dissolves the aforementioned difficulty in solving the least-squares solution. Finally, by applying the Moore-Penrose generalized inverse and Kronecker product of matrices, the least squares $\eta$-Hermitian solution and unique minimal norm solution of the split quaternion matrix equation are deduced. The feasibility of the proposed approach is verified by the numerical example. Related Articles | Metrics

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The Convergence of Generalized Non-stationary Multi-splitting Two-stage Iterative Methods for Semi-definite Linear Systems CUI Yanxing, WANG Chuanlong, JIANG Wensheng 2022, 39 (5):  826-834.  doi: 10.3969/j.issn.1005-3085.2022.05.011 Abstract ( 70 )   PDF (181KB) ( 63 )   Save To effectively solve the large sparse linear equations of positive definite or positive semidefinite, the second stage splitting of generalized non-stationary multi-splitting two-stage iterative method is proposed, which combines the techniques of multi-splitting and matrix preprocessing generalizes the non-stationary multi-splitting two-stage iterative method based on the classical matrix splitting in the first stage. The algorithm and logical expression of the generalized non-stationary multi-stage iterative method are rewritten into a compact iterative scheme to consider the convergence. According to the iterative scheme, the generalized non-stationary multi-splitting two-stage iterative algorithm is convergent under a sufficient condition. Finally, a numerical example of a large sparse linear system with a five-diagonal coefficient matrix shows the feasibility of the generalized non-stationary multi-split two-stage iterative algorithm, and the efficiency of the algorithm is verified in terms of iteration steps and CPU time. Related Articles | Metrics

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Dynamic Evolution Analysis of the New Financial Duffing-Holms Model and Its Control ZHAO Yue, XU Yuhua, XIE Chengrong 2022, 39 (5):  835-844.  doi: 10.3969/j.issn.1005-3085.2022.05.012 Abstract ( 92 )   PDF (4284KB) ( 74 )   Save In this paper, a new financial Duffing-Holms chaotic model is proposed, and the basic dynamical properties of the system are discussed, such as Hopf bifurcation, dissipation, Lyapunov exponent, Poincar\'{e} diagram and bifurcation diagram. A new finite-time convergence theorem is proposed for general chaotic systems. Compared with the existing finite-time control of chaotic systems, the new finite-time controller with fractional index greater than 1 can also realize the finite-time synchronization of financial Duffing-Holms chaotic systems. The validity of the theoretical results is verified by numerical simulation. Related Articles | Metrics

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Bifurcation and Stability Analysis of a Fractional-order Shape Memory Alloy Oscillator with Delayed Feedback WANG Jinbin, MA Lifeng 2022, 39 (5):  845-850.  doi: 10.3969/j.issn.1005-3085.2022.05.013 Abstract ( 56 )   PDF (261KB) ( 202 )   Save The fractional order is introduced into the shape memory alloy oscillator model, and the effect of fractional order on the system dynamics is discussed. Firstly, based on the fractional differential equation theory, the fractional shape memory alloy system (FSMA) is constructed, and the stability of the system and the existence conditions of Hopf bifurcation are presented. Then, a time-delay feedback controller is designed to control the stability of the FSMA system. Finally, the numerical results indicate that the time delay and fractional order play important roles in regulating the dynamic properties of the shape memory alloy system. Related Articles | Metrics