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A LINEARLY-FITTED CONSERVATIVE （DISSIPATIVE） SCHEME FOR EFFICIENTLY SOLVING CONSERVATIVE （DISSIPATIVE） NONLINEAR WAVE PDES

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Journal of Computational Mathematics 2017年第6期35卷 780-800页

作者： Kai Liu XinyuanWu Wei Shi College of Applied Mathematics Nanjing University of Finance ~ Economics Nanjing 210023 China Department of Mathematics Nanjing University State Key Laboratory for Novel Software Technology at Nanjing University Nanjing 210093 China College of Mathematical Sciences Nanjing Teeh University Nanjing 211816 China

The extended discrete gradient method is an extension of traditional discrete gradient method, which is specially designed to solve oscillatory Hamiltonian systems efficiently while preserving their energy exactly. In this paper, based on the extended discrete gradient method, we present an efficient approach to devising novel schemes for numerically solving conservative （dissipative） nonlinear wave partial differential equations. The new scheme can preserve the energy exactly for conservative wave equations. With a minor remedy to the extended discrete gradient method, the new scheme is applicable to dissipative wave equations. Moreover, it can preserve the dissipation structure for the dissipative wave equation as well. Another important property of the new scheme is that it is linearly-fitted, which guarantees much fast convergence for the fixed-point iteration which is required by an energy-preserving integrator. The efficiency of the new scheme is demonstrated by some numerical examples.

关键词： Conservative （dissipative） wave PDEs structure-preserving algorithm Linearly-fitted Average Vector Field formula Sine-Gordon equation.

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Comments on 'A structure-preserving method for the quaternion LU decomposition in quaternionic quantum theory' by Minghui Wang and Wenhao Ma

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COMPUTER PHYSICS COMMUNICATIONS 2015年 188卷 128-130页

作者： Sangwine, Stephen J. Univ Essex Sch Comp Sci & Elect Engn Colchester CO4 3SQ Essex England

Claims were made in an article by Wang and Ma in 2013 that they had devised an algorithm for the quaternion LU decomposition that was significantly faster than the LU decomposition implemented in the Quaternion Toolbox for MATLAB (QTFM). These claims have been tested and found to be unsupported by MATLAB code supplied to the author by Wang and Ma. The author's tests are presented, and test code made available as supplementary material. It is found that not only is the QTFM code faster, but that Wang and Ma's algorithm has run-time that scales with the square of the size of the matrix, whereas the algorithm in QTFM has run time approximately linear in matrix size. These findings are consistent with an inspection of the code. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Quaternion matrix LU decomposition structure-preserving algorithm

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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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A structure-preserving iteration method of model updating based on matrix approximation theory

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COMPUTATIONAL & APPLIED MATHEMATICS 2010年第2期29卷 173-194页

作者： Xie, Dongxiu Beijing Informat Sci & Technol Univ Sch Sci Beijing 100192 Peoples R China

Some theories and a method are discussed on updating a generalized centrosymmetric model. It gives a generalized centrosymmetric modified solution with partial prescribed least square spectra constraints. The emphasis is given on exploiting structure-preserving algorithm based on matrix approximation theory. A perturbation theory for the modified solution is established. The convergence of an iterative solution is investigated. Illustrative examples are provided.

关键词： structure-preserving algorithm generalized centrosymmetric matrix model updating perturbation analysis

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CP DECOMPOSITION AND LOW-RANK APPROXIMATION OF ANTISYMMETRIC TENSORS

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ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS 2024年 62卷 72-94页

作者： Kovac, Erna Begovic Perisa, Lana Univ Zagreb Fac Chem Engn & Technol Marulicev Trg 19 Zagreb 10000 Croatia Visage Technol Ivana Lucica 2a Zagreb 10000 Croatia

For antisymmetric tensors, the paper examines a low-rank approximation that is represented via only three vectors. We describe a suitable low-rank format and propose an alternating least-squares structure-preserving algorithm for finding such an approximation. Moreover, we show that this approximation problem is equivalent to the problem of finding the best multilinear low-rank antisymmetric approximation and, consequently, equivalent to the problem of finding the best unstructured rank-1 approximation. The case of partial antisymmetry is also discussed. The algorithms are implemented in the Julia programming language and their numerical performance is discussed.

关键词： CP decomposition antisymmetric tensors low-rank approximation structure-preserving algorithm Julia

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A linearly implicit energy-preserving exponential time differencing scheme for the fractional nonlinear Schrodinger equation

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NETWORKS AND HETEROGENEOUS MEDIA 2023年第3期18卷 1105-1117页

作者： Ma, Tingting Fu, Yayun He, Yuehua Yang, Wenjie Zhoukou Normal Univ Zhoukou 466000 Peoples R China Xuchang Univ Sch Sci Xuchang 461000 Peoples R China

In this paper, we present a new method to solve the fractional nonlinear Schrodinger equation. Our approach combines the invariant energy quadratization method with the exponential time differencing method, resulting in a linearly-implicit energy-preserving scheme. To achieve this, we introduce an auxiliary variable to derive an equivalent system with a modified energy conservation law. The proposed scheme uses stabilized exponential time differencing approximations for time integration and Fourier pseudo-spectral discretization in space to obtain a linearly-implicit, fully-discrete scheme. Compared to the original energy-preserving exponential integrator scheme, our approach is more efficient as it does not require nonlinear iterations. Numerical experiments confirm the effectiveness of our scheme in conserving energy and its efficiency in long-time computations.

关键词： fractional nonlinear Schrodinger equation invariant energy quadratization structure-preserving algorithm exponential time differencing

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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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Efficient invariant-preserving scheme for the N-coupled nonlinear Schrödinger equations

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APPLIED MATHEMATICS LETTERS 2024年 157卷

作者： Cai, Jiaxiang Huaiyin Normal Univ Sch Math Sci Huaian 223300 Jiangsu Peoples R China Minist Educ Key Lab NSLSCS Nanjing 210023 Jiangsu Peoples R China

The N -coupled nonlinear Schr & ouml;dinger equations are reformulated into an expanded form by using a scalar auxiliary variable method, and then a scheme preserving exactly original mass and energy conservation laws is proposed based on discretizing the expanded form. The scheme is efficient as it consists of decoupled linear systems with constant coefficients, along with a nonlinear algebraic equation that can be solved with negligible computational cost. Some numerical experiments are carried out to demonstrate the behavior of wave solutions, the accuracy of solution and the preservation of physical invariants.

关键词： Schr & ouml dinger equation structure-preserving algorithm Scalar auxiliary variable Energy-preserving algorithm Decoupled scheme

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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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