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An efficient energy-preserving method for the two-dimensional fractional Schrodinger equation

APPLIED NUMERICAL MATHEMATICS 2021年 165卷 232-247页

作者： Fu, Yayun Xu, Zhuangzhi Cai, Wenjun Wang, Yushun Xuchang Univ Sch Sci Xuchang 461000 Peoples R China Nanjing Normal Univ Jiangsu Collaborat Innovat Ctr Biomedial Funct Ma Sch Math Sci Jiangsu Key Lab NSLSCS Nanjing 210023 Peoples R China

In this paper, we study the Hamiltonian structure and develop a novel energy-preserving scheme for the two-dimensional fractional nonlinear Schrodinger equation. First, we present the variational derivative of the functional with fractional Laplacian to derive the Hamiltonian formula of the equation and obtain an equivalent system by defining a scalar variable. An energy-preserving scheme is then presented by applying exponential time differencing approximations for time integration and Fourier pseudo-spectral discretization in space. The proposed scheme is a linear system and can be solved efficiently. Numerical experiments are displayed to verify the conservation, efficiency, and good performance at a relatively large time step in long time computations. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Fractional nonlinear Schrodinger equation Hamiltonian structure structure-preserving algorithm Exponential time differencing

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A linearly implicit structure-preserving scheme for the fractional sine-Gordon equation based on the IEQ approach

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APPLIED NUMERICAL MATHEMATICS 2021年 160卷 368-385页

作者： Fu, Yayun Cai, Wenjun Wang, Yushun Xuchang Univ Sch Sci Xuchang 461000 Peoples R China Nanjing Normal Univ Jiangsu Collaborat Innovat Ctr Biomedial Funct Ma Sch Math Sci Jiangsu Key Lab NSLSCS Nanjing 210023 Peoples R China

This paper aims to develop a linearly implicit structure-preserving numerical scheme for the space fractional sine-Gordon equation, which is based on the newly developed invariant energy quadratization method. First, we reformulate the equation as a canonical Hamiltonian system by virtue of the variational derivative of the functional with fractional Laplacian. Then, we utilize the fractional centered difference formula to discrete the equivalent system derived by the invariant energy quadratization method in space direction, and obtain a conservative semi-discrete scheme. Subsequently, the linearly implicit structure-preserving method is applied for the resulting semi-discrete system to arrive at a fully-discrete conservative scheme. The stability, solvability and convergence in the maximum norm of the numerical scheme are given. Furthermore, a fast algorithm based on the fast Fourier transformation technique is used to reduce the computational complexity in practical computation. Finally, numerical examples are provided to confirm our theoretical analysis results. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： structure-preserving algorithm Fractional sine-Gordon equation Hamiltonian system Invariant energy quadratization Numerical analysis

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A dissipation-preserving scheme for damped oscillatory Hamiltonian systems based on splitting

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APPLIED NUMERICAL MATHEMATICS 2021年 170卷 242-254页

作者： Liu, Kai Fu, Ting Shi, Wei Nanjing Univ Finance & Econ Coll Appl Math Nanjing 210023 Peoples R China Nanjing Tech Univ Coll Math Sci Nanjing 211816 Peoples R China

In this paper, a new dissipation-preserving scheme is established for weakly dissipative perturbations of oscillatory Hamiltonian systems. The system exhibits a nonlinear oscillatory structure. The main oscillation is governed by a matrix M and the damping is governed by a matrix Gamma. The new scheme preserves the oscillatory structure of the systems by incorporating the matrix M in the scheme based on the idea of ERKN methods. Meanwhile, the discrete gradient and splitting are used to construct the scheme such that the numerical solution possesses a nearly correct damping rate of the system. A main feature of the new scheme is that a relatively large stepsize can be chosen since the convergence of the implicit iterations in the scheme is shown to be independent of the matrices M and Gamma. Three numerical experiments of perturbed Hamiltonian systems are conducted to show the effectiveness and the efficiency of the new scheme in comparison with the traditional discrete gradient methods. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Weakly dissipative Hamiltonian systems Oscillatory systems structure-preserving algorithm Discrete gradient Sine-Gordon equation Continuous Fermi-Pasta-Ulam system

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Learnability of linear port-hamiltonian systems

The Journal of Machine Learning Research

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The Journal of Machine Learning Research 2024年第1期25卷 3284-3339页

作者： Juan-Pablo Ortega Daiying Yin Division of Mathematical Sciences Nanyang Technological University Singapore

A complete structure-preserving learning scheme for single-input/single-output (SISO) linear port-Hamiltonian systems is proposed. The construction is based on the solution, when possible, of the unique identification problem for these systems, in ways that reveal fundamental relationships between classical notions in control theory and crucial properties in the machine learning context, like structure-preservation and expressive power. In the canonical case, it is shown that, up to initializations, the set of uniquely identified systems can be explicitly characterized as a smooth manifold endowed with global Euclidean coordinates, which allows concluding that the parameter complexity necessary for the replication of the dynamics is only O(n) and not O(n2), as suggested by the standard parametrization of these systems. Furthermore, it is shown that linear port-Hamiltonian systems can be learned while remaining agnostic about the dimension of the underlying data-generating system. Numerical experiments show that this methodology can be used to efficiently estimate linear port-Hamiltonian systems out of input-output realizations, making the contributions in this paper the first example of a structure-preserving machine learning paradigm for linear port-Hamiltonian systems based on explicit representations of this model category.

关键词： linear port-Hamiltonian system machine learning structure-preserving algorithm systems theory physics-informed machine learning unique identification problem controllable representation observable representation canonical representation

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Local structure-preserving algorithms for general multi-symplectic Hamiltonian PDEs

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COMPUTER PHYSICS COMMUNICATIONS 2019年 235卷 210-220页

作者： Cai, Jiaxiang Wang, Yushun Jiang, Chaolong Huaiyin Normal Univ Sch Math Sci Huaian 223300 Jiangsu Peoples R China Nanjing Normal Univ Jiangsu Key Lab NSLSCS Sch Math Sci Nanjing 210046 Jiangsu Peoples R China

Many PDEs can be recast into the general multi-symplectic formulation possessing three local conservation laws. We devote the present paper to some systematic methods, which hold the discrete versions of the local conservation laws respectively, for the general multi-symplectic PDEs. For the original problem subjected to appropriate boundary conditions, the proposed methods are globally conservative. The proposed methods are successfully applied to many one-dimensional and multi-dimensional Hamiltonian PDEs, such as KdV equation, G-P equation, Maxwell's equations and so on. Numerical experiments are carried out to verify the theoretical analysis. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Multi-symplectic Hamiltonian PDE structure-preserving algorithm Conservation law Discrete gradient Dispersion relation

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Unconditional Convergence of Conservative Spectral Galerkin Methods for the Coupled Fractional Nonlinear Klein-Gordon-Schrodinger Equations

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JOURNAL OF SCIENTIFIC COMPUTING 2023年第3期94卷 70-70页

作者： Hu, Dongdong Fu, Yayun Cai, Wenjun Wang, Yushun Jiangxi Normal Univ Jiangxi Prov Ctr Appl Math Nanchang 330022 Peoples R China Jiangxi Normal Univ Sch Math & Stat Nanchang 330022 Peoples R China Xuchang Univ Sch Sci Xuchang 461000 Peoples R China Nanjing Normal Univ Sch Math Sci Nanjing 210023 Peoples R China

In this work, two novel classes of structure-preserving spectral Galerkin methods are proposed which based on the Crank-Nicolson scheme and the exponential scalar auxiliary variable method respectively, for solving the coupled fractional nonlinear Klein-Gordon-Schrodinger equation. The paper focuses on the theoretical analyses and computational efficiency of the proposed schemes, the Crank-Nicoloson scheme is proved to be unconditionally convergent and has maximum-norm boundness of numerical solutions. The exponential scalar auxiliary variable scheme is linearly implicit and decoupled, but lack of the maximum-norm boundness, also, the energy structure is modified. Subsequently, the efficient implementations of the proposed schemes are introduced in detail. Both the theoretical analyses and the numerical comparisons show that the proposed spectral Galerkin methods have high efficiency in long-time computations.

关键词： Riesz fractional derivative Spectral Galerkin method structure-preserving algorithm Unique solvability Convergence

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Joint diagonalization for a pair of Hermitian quaternion matrices and applications to color face recognition

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SIGNAL PROCESSING 2022年 198卷

作者： Ling, Si-Tao Li, Yi-Ding Yang, Bing Jia, Zhi-Gang China Univ Min & Technol Sch Math Xuzhou 221116 Peoples R China Jiangsu Normal Univ Sch Math & Stat Xuzhou 221116 Peoples R China Jiangsu Normal Univ Res Inst Math Sci Xuzhou 221116 Peoples R China

A new joint diagonalization algorithm for a pair of Hermitian quaternion matrices is derived incorporating real structure-preserving strategy. The structure-preserving joint diagonalization algorithm leads to a novel two-dimensional quaternion linear discriminant analysis (2D-QLDA) method for color face recognition and image reconstruction. 2D-QLDA is mathematically characterized by Hermitian quaternion generalized eigenvalue problem. A weighted norm is obtained as a new measurement to determine the distances among Fisher feature matrices, which helps us avoid generating projected images explicitly. Numerical results based on the real face databases indicate that 2D-QLDA performs better than other 2D-LDA-like methods in color face recognition and is effective in image reconstruction. (c) 2022 Elsevier B.V. All rights reserved.

关键词： Hermitian quaternion matrix structure-preserving algorithm Joint diagonalization Face recognition Linear discriminant analysis

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An explicitly solvable energy-conserving algorithm for pitch-angle scattering in magnetized plasmas

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JOURNAL OF COMPUTATIONAL PHYSICS 2022年 449卷 110767-110767页

作者： Fu, Yichen Zhang, Xin Qin, Hong Princeton Univ Princeton NJ USA Princeton Plasma Phys Lab Princeton NJ USA Princeton Univ Dept Astrophys Sci Princeton NJ 08544 USA

We develop an Explicitly Solvable Energy-Conserving (ESEC) algorithm for the Stochastic Differential Equation (SDE) describing the pitch-angle scattering process in magnetized plasmas. The Cayley transform is used to calculate both the deterministic gyromotion and stochastic scattering, affording the algorithm to be explicitly solvable and exactly energy conserving. An unusual property of the SDE for pitch-angle scattering is that its coefficients diverge at the zero velocity and do not satisfy the global Lipschitz condition. Consequently, when standard numerical methods, such as the Euler-Maruyama (EM), are applied, numerical convergence is difficult to establish. For the proposed ESEC algorithm, its energy-preserving property enables us to overcome this obstacle. We rigorously prove that the ESEC algorithm is order 1/2 strongly convergent. This result is confirmed by detailed numerical studies. For the case of pitch-angle scattering in a magnetized plasma with a constant magnetic field, the numerical solution is benchmarked against the analytical solution, and excellent agreements are found. (C) 2021 Elsevier Inc. All rights reserved.

关键词： Plasmas Collisions Stochastic differential equation structure-preserving algorithm Monte Carlo method

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A double-color image watermarking algorithm based on quaternion Schur decomposition

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OPTIK 2022年 269卷

作者： Sun, Yehan Su, Qingtang Chen, Siyu Zhang, Xueting Ludong Univ Sch Informat & Elect Engn Yantai 264025 Peoples R China

In view of the fact that many watermarking algorithms used for copyright protection have ignored the correlation and synchronization between the color channels, a new double-color image watermarking algorithm based on quaternion Schur (QSchur) decomposition is presented in this paper. Firstly, the color watermark image is encrypted by two-dimensional compound chaotic map;then, the QSchur decomposition is used to associate and transform multiple color channels;finally, the watermark is embedded and extracted blindly by using the high correlation between the coefficient pairs in the block. Due to the high time complexity of the quaternion, a real structure-preserving algorithm is used to shorten the running time of the proposed algorithm by about 20 times compared with the quaternion toolbox of Matlab. Simulation experiments and analysis demonstrate that this algorithm has stronger robustness and higher security when the actual embedding rate and invisibility are the same as those of other algorithms given.

关键词： Watermarking Quaternion Schur decomposition structure-preserving algorithm Compound chaotic map

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Hamiltonian Particle-in-Cell methods for Vlasov-Poisson equations

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JOURNAL OF COMPUTATIONAL PHYSICS 2022年 467卷 1页

作者： Gu, Anjiao He, Yang Sun, Yajuan Chinese Acad Sci Acad Math & Syst Sci ICMSEC LSEC Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing 100049 Peoples R China Univ Sci & Technol Beijing Sch Math & Phys Beijing 100083 Peoples R China

In this paper, Particle-in-Cell algorithms for the Vlasov-Poisson system are presented based on its Poisson bracket structure. The Poisson equation is solved by finite element methods, in which the appropriate finite element spaces are taken to guarantee that the semi-discretized system possesses a well defined discrete Poisson bracket structure. Then, splitting methods are applied to the semi-discretized system by decomposing the Hamiltonian function. The resulting discretizations are proved to be Poisson bracket preserving. Moreover, the conservative quantities of the system are also well preserved. In numerical experiments, we use the presented numerical methods to simulate various physical phenomena. Due to the huge computational effort of the practical computations, we employ the strategy of parallel computing. The numerical results verify the efficiency of the new derived numerical discretizations. (C) 2022 Elsevier Inc. All rights reserved.

关键词： Vlasov-Poisson system Poisson bracket Finite element method structure-preserving algorithm Hamiltonian splitting method

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