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QR decomposition of dual quaternion matrix and blind watermarking scheme 双四元矩阵的 QR 分解和盲水印方案
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-09-13 DOI: 10.1007/s11075-024-01930-9
Mingcui Zhang, Ying Li, Tao Wang, Jianhua Sun

In this paper, the algorithms and applications of the dual quaternion QR decomposition are studied. The direct algorithm and dual structure-preserving algorithm of dual quaternion QR decomposition utilizing Householder transformation of dual quaternion vector are proposed. Numerical experiments show that two algorithms are feasible, and the dual structure-preserving algorithm is superior to the direct algorithm in terms of computational efficiency. Therefore, the dual structure-preserving algorithm of dual quaternion QR decomposition is used to color image watermarking. Experiments illustrate that our method is feasible and better than the compared methods in anti-aggression.

本文研究了对偶四元数 QR 分解的算法和应用。本文提出了利用双四元数矢量的 Householder 变换进行双四元数 QR 分解的直接算法和双结构保留算法。数值实验表明,两种算法都是可行的,而且就计算效率而言,双结构保留算法优于直接算法。因此,双四元 QR 分解的双结构保留算法被用于彩色图像水印。实验表明,我们的方法是可行的,而且在抗攻击性方面优于其他方法。
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引用次数: 0
An effective real structure-preserving algorithm for the quaternion indefinite least squares problem 四元不定最小二乘问题的有效实结构保留算法
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-09-12 DOI: 10.1007/s11075-024-01929-2
Zixiang Meng, Zhihan Zhou, Ying Li, Fengxia Zhang

This paper concentrates on the quaternion indefinite least squares (QILS) problem. Firstly, we define the quaternion J-unitary matrix and the quaternion hyperbolic Givens rotation, and study their properties. Then, based on these, we investigate the quaternion hyperbolic QR factorization, and purpose its real structure-preserving (SP) algorithm by the real representation (Q-RR) matrix of the quaternion matrix. Immediately after, we explore the solution of the QILS problem, and give a real SP algorithm of solving the QILS problem. Eventually, to illustrate the effectiveness of proposed algorithms, we offer numerical examples.

本文主要研究四元数不定最小二乘(QILS)问题。首先,我们定义了四元数 J 单位矩阵和四元数双曲 Givens 旋转,并研究了它们的性质。在此基础上,我们研究了四元双曲 QR 因式分解,并通过四元矩阵的实表示(Q-RR)矩阵实现了其实结构保留(SP)算法。紧接着,我们探讨了 QILS 问题的求解,并给出了求解 QILS 问题的实 SP 算法。最后,为了说明所提算法的有效性,我们提供了数值示例。
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引用次数: 0
A trust-region framework for iteration solution of the direct INDSCAL problem in metric multidimensional scaling 度量多维标度中直接 INDSCAL 问题迭代求解的信任区域框架
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-09-11 DOI: 10.1007/s11075-024-01921-w
Xue-lin Zhou, Chao-qian Li

The well-known INdividual Differences SCALing (INDSCAL) model is intended for the simultaneous metric multidimensional scaling (MDS) of several doubly centered matrices of squared dissimilarities. An alternative approach, called for short DINDSCAL (direct INDSCAL), is proposed for analyzing directly the input matrices of squared dissimilarities. In the present work, the problem of fitting the DINDSCAL model to the data is formulated as a Riemannian optimization problem on a product matrix manifold comprised of the Stiefel sub-manifold of zero-sum matrices and non-negative diagonal matrices. A practical algorithm, based on the generic Riemannian trust-region method by Absil et al., is presented to address the underlying problem, which is characterized by global convergence and local superlinear convergence rate. Numerical experiments are conducted to illustrate the efficiency of the proposed method. Furthermore, comparisons with the existing projected gradient approach and some classical methods in the MATLAB toolbox Manopt are also provided to demonstrate the merits of the proposed approach.

众所周知的 INdividual Differences SCALing(INDSCAL)模型用于同时对多个双中心异同平方矩阵进行度量多维标度(MDS)。我们提出了另一种方法,简称为 DINDSCAL(直接 INDSCAL),用于直接分析输入的差异平方矩阵。在本研究中,DINDSCAL 模型与数据的拟合问题被表述为一个乘积矩阵流形上的黎曼优化问题,乘积矩阵流形由零和矩阵和非负对角矩阵的 Stiefel 子流形组成。在 Absil 等人提出的通用黎曼信任区域法基础上,提出了一种实用算法来解决基本问题,该算法具有全局收敛性和局部超线性收敛率的特点。通过数值实验说明了所提方法的效率。此外,还与现有的投影梯度法和 MATLAB 工具箱 Manopt 中的一些经典方法进行了比较,以证明所提方法的优点。
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引用次数: 0
A numerical algorithm for the computation of the noncentral beta distribution function 计算非中心贝塔分布函数的数值算法
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-09-10 DOI: 10.1007/s11075-024-01931-8
Vera Egorova, Amparo Gil, Javier Segura, Nico M. Temme

The noncentral beta distribution function is a generalization of the central beta distribution (the regularized incomplete beta function) that includes a noncentrality parameter. This paper describes an algorithm and provides a Matlab implementation for calculating the noncentral beta distribution function. Through a series of numerical tests, we demonstrate that the algorithm is accurate and efficient across a wide range of parameters.

非中心贝塔分布函数是中心贝塔分布(正则化不完全贝塔函数)的广义化,包含一个非中心性参数。本文介绍了计算非中心贝塔分布函数的算法并提供了 Matlab 实现。通过一系列数值测试,我们证明了该算法在广泛的参数范围内都是准确高效的。
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引用次数: 0
Numerical algorithms for recovering a fractional Sturm-Liouville operator based on trace formulae 基于迹公式的分数斯特姆-利乌维尔算子恢复数值算法
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-09-07 DOI: 10.1007/s11075-024-01926-5
Xiaowen Li, Xiaoying Jiang, Xiang Xu

This paper explores numerical methods for recovering a density term in a fractional Sturm-Liouville problem using a set of spectra. By applying Lidskii’s theorem, a sequence of trace formulae are derived to elucidate the connections between the unknown coefficients and the complex eigenvalues of a fractional spectra problem. Two efficient algorithms are proposed based on these trace formulae, and their effectiveness is demonstrated through numerical experiments.

本文探讨了利用一组谱恢复分数 Sturm-Liouville 问题中的密度项的数值方法。通过应用 Lidskii 定理,推导出一系列迹公式,以阐明未知系数与分数谱问题复特征值之间的联系。根据这些迹公式提出了两种高效算法,并通过数值实验证明了它们的有效性。
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引用次数: 0
Convergence of a partially truncated Euler-Maruyama method for SDEs with super-linear piecewise continuous drift and Hölder diffusion coefficients 具有超线性片断连续漂移和霍尔德扩散系数的 SDE 的部分截断欧拉-Maruyama 方法的收敛性
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-09-04 DOI: 10.1007/s11075-024-01928-3
Amir Haghighi

The main purpose of this paper is to develop and analyze a partially truncated Euler-Maruyama method for numerically solving SDEs with super-linear piecewise continuous drift coefficients and (varvec{(1/2+alpha )})-Hölder diffusion coefficients (PTEMH), for (varvec{alpha in [0,1/2]}). We first present an analytical form for the unique solution of such problems. Then we establish the strong convergence theory of the PTEMH scheme. We show that the convergence rate of the proposed method in the case (varvec{alpha in (0,1/2]}) reaches (varvec{alpha }), which is optimal compared to the explicit Euler-Maruyama method. Finally, numerical results are given to confirm the theoretical convergence rate.

本文的主要目的是开发和分析一种部分截断的 Euler-Maruyama 方法,用于数值求解具有超线性片断连续漂移系数和 (varvec{(1/2+alpha )})-Hölder 扩散系数(PTEMH)的 SDEs,对于 (varvec{alpha in [0,1/2]}).我们首先给出了此类问题唯一解的解析形式。然后,我们建立了 PTEMH 方案的强收敛理论。我们表明,在 (varvec{alpha in (0,1/2]}) 的情况下,所提出方法的收敛速率达到了 (varvec{alpha }) ,这与显式 Euler-Maruyama 方法相比是最优的。最后,给出的数值结果证实了理论上的收敛速度。
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引用次数: 0
The fast Euler-Maruyama method for solving multiterm Caputo fractional stochastic delay integro-differential equations 求解多期卡普托分数随机延迟积分微分方程的快速欧拉-丸山方法
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-09-03 DOI: 10.1007/s11075-024-01925-6
Huijiao Guo, Jin Huang, Yi Yang, Xueli Zhang

This paper studies a type of multiterm fractional stochastic delay integro-differential equations (FSDIDEs). First, the Euler-Maruyama (EM) method is developed for solving the equations, and the strong convergence order of this method is obtained, which is (varvec{min left{ alpha _{l}-frac{1}{2}, alpha _{l}-alpha _{l-1}right} }). Then, a fast EM method is also presented based on the exponential-sum-approximation with trapezoid rule to cut down the computational cost of the EM method. In the end, some concrete numerical experiments are used to substantiate these theoretical results and show the effectiveness of the fast method.

本文研究了一种多期分数随机延迟积分微分方程(FSDIDEs)。首先,建立了求解该方程的 Euler-Maruyama (EM) 方法,并得到了该方法的强收敛阶数,即 (varvec{min left{ alpha _{l}-frac{1}{2}, alpha _{l}-alpha _{l-1}right} 。}).然后,还提出了一种基于梯形法则的指数和逼近的快速 EM 方法,以降低 EM 方法的计算成本。最后,通过一些具体的数值实验来证实这些理论结果,并展示了快速方法的有效性。
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引用次数: 0
Analysis of a higher-order scheme for multi-term time-fractional integro-partial differential equations with multi-term weakly singular kernels 具有多期弱奇异内核的多期时分整偏微分方程的高阶方案分析
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-09-03 DOI: 10.1007/s11075-024-01927-4
Sudarshan Santra

This work is focused on developing a hybrid numerical method that combines a higher-order finite difference method and multi-dimensional Hermite wavelets to address two-dimensional multi-term time-fractional integro-partial differential equations with multi-term weakly singular kernels having bounded and unbounded time derivatives at the initial time (t=0). Specifically, the multi-term fractional operators are discretized using a higher-order approximation designed by employing different interpolation schemes based on linear, quadratic, and cubic interpolation leading to (mathcal {O}(N^{-(4-alpha _1)})) accuracy on a suitably chosen nonuniform mesh and (mathcal {O}(N^{-alpha _1})) accuracy on a uniformly distributed mesh. The weakly singular integral operators are approximated by a modified numerical quadrature, which is a combination of the composite trapezoidal approximation and the midpoint rule. The effects of the exponents of the weakly singular kernels over fractional orders are analyzed in terms of accuracy over uniform and nonuniform meshes for the solution having both bounded and unbounded time derivatives. The stability of the proposed semi-discrete scheme is derived based on (L^infty )-norm for uniformly distributed temporal mesh. Further, we employ the uniformly distributed collocation points in spatial directions to estimate the tensor-based wavelet coefficients. Moreover, the convergence analysis of the fully discrete scheme is carried out based on (L^2)-norm leading to (mathcal {O}(N^{-alpha _1})) accuracy on a uniform mesh. It also highlights the higher-order accuracy over nonuniform mesh. Additionally, we discuss the convergence analysis of the proposed scheme in the context of the multi-term time-fractional diffusion equations involving time singularity demonstrating a (mathcal {O}(N^{-(4-alpha _1)})) accuracy on a nonuniform mesh with suitably chosen grading parameter. Note that the scheme reduces to (mathcal {O}(N^{-alpha _1})) accuracy on a uniform mesh. Several tests are performed on numerous examples in (L^infty )- and (L^2)-norm to show the efficiency of the proposed method. Further, the solutions’ nature and accuracy in terms of absolute point-wise error are illustrated through several isosurface plots for different regularities of the exact solution. These experiments confirm the theoretical accuracy and guarantee the convergence of approximations to the functions having time singularity, and the higher-order accuracy for a suitably chosen nonuniform mesh.

这项工作的重点是开发一种混合数值方法,该方法结合了高阶有限差分法和多维赫米特小波,用于处理二维多期时间分式整偏微分方程,该方程具有多期弱奇异核,在初始时间具有有界和无界时间导数(t=0)。具体地说,多期分式算子通过采用基于线性、二次和三次插值的不同插值方案进行高阶近似离散化,从而在适当选择的非均匀网格上达到(mathcal {O}(N^{-(4-alpha _1)}))精度,在均匀分布网格上达到(mathcal {O}(N^{-alpha _1}))精度。弱奇异积分算子采用修正的数值正交近似,它是复合梯形近似和中点规则的结合。对于有界和无界时间导数的解,从均匀和非均匀网格的精度角度分析了弱奇异核指数对分数阶的影响。基于均匀分布时间网格的 (L^infty )-norm,得出了所提出的半离散方案的稳定性。此外,我们利用空间方向上均匀分布的定位点来估计基于张量的小波系数。此外,基于 (L^2)-norm 对完全离散方案进行了收敛分析,从而在均匀网格上达到了 (mathcal {O}(N^{-alpha _1}))精度。它还强调了在非均匀网格上的高阶精度。此外,我们还讨论了在涉及时间奇异性的多期时间-分数扩散方程背景下所提出方案的收敛性分析,证明了在适当选择分级参数的非均匀网格上的(mathcal {O}(N^{-(4-alpha _1)}))精度。请注意,该方案在均匀网格上的精度可降低到 (mathcal {O}(N^{-alpha _1})/)。在 (L^infty )-和 (L^2) -规范下对大量实例进行了测试,以显示所提方法的效率。此外,通过精确解的不同规则性的等值面图,说明了解的性质和绝对点误差的准确性。这些实验证实了理论上的准确性,并保证了具有时间奇异性的函数近似值的收敛性,以及在适当选择非均匀网格时的高阶准确性。
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引用次数: 0
VEMcomp: a Virtual Elements MATLAB package for bulk-surface PDEs in 2D and 3D VEMcomp:用于二维和三维体面 PDE 的虚拟元素 MATLAB 软件包
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-08-31 DOI: 10.1007/s11075-024-01919-4
Massimo Frittelli, Anotida Madzvamuse, Ivonne Sgura

We present a Virtual Element MATLAB solver for elliptic and parabolic, linear and semilinear Partial Differential Equations (PDEs) in two and three space dimensions, which is coined VEMcomp. Such PDEs are widely applicable to describing problems in material sciences, engineering, cellular and developmental biology, among many other applications. The library covers linear and nonlinear models posed on different simple and complex geometries, involving time-dependent bulk, surface, and bulk-surface PDEs. The solver employs the Virtual Element Method (VEM) of lowest polynomial order ({k=1}) on general polygonal and polyhedral meshes, including the Finite Element Method (FEM) of order ({k=1}) as a special case when the considered mesh is simplicial. VEMcomp has three main purposes. First, VEMcomp generates polygonal and polyhedral meshes optimized for fast matrix assembly. Triangular and tetrahedral meshes are encompassed as special cases. For surface PDEs, VEMcomp is compatible with the well-known Matlab package DistMesh for mesh generation. Second, given a mesh for the considered geometry, possibly generated with an external package, VEMcomp computes all the matrices (mass and stiffness) required by the VEM or FEM method. Third, for multiple classes of stationary and time-dependent bulk, surface and bulk-surface PDEs, VEMcomp solves the considered PDE problem with the VEM or FEM in space and IMEX Euler in time, through a user-friendly interface. As an optional post-processing, VEMcomp comes with its own functions for plotting the numerical solutions and evaluating the error when possible. An extensive set of examples illustrates the usage of the library.

我们提出了一个虚拟元素 MATLAB 求解器,用于求解二维和三维空间的椭圆和抛物线、线性和半线性偏微分方程(PDEs),并将其命名为 VEMcomp。这类偏微分方程广泛应用于描述材料科学、工程学、细胞和发育生物学等领域的问题。该库涵盖在不同的简单和复杂几何形状上建立的线性和非线性模型,涉及与时间相关的体动力学、表面动力学和体-面 PDEs。该求解器在一般多边形和多面体网格上采用了最低多项式阶数 ({k=1}) 的虚拟元素法(VEM),包括阶数 ({k=1}) 的有限元法(FEM),作为当考虑的网格是简单网格时的一种特殊情况。VEMcomp 有三个主要用途。首先,VEMcomp 生成的多边形和多面体网格经过优化,可用于快速矩阵组装。三角形和四面体网格作为特例也包括在内。对于曲面 PDE,VEMcomp 与著名的 Matlab 网格生成软件包 DistMesh 兼容。其次,VEMcomp 会根据所考虑几何体的网格(可能由外部软件包生成)计算 VEM 或 FEM 方法所需的所有矩阵(质量和刚度)。第三,对于多类静态和时间相关的体动力学、表面动力学和体-表面动力学问题,VEMcomp 可通过用户友好界面,使用空间 VEM 或 FEM 和时间 IMEX Euler 解决所考虑的 PDE 问题。作为可选的后处理功能,VEMcomp 自带绘制数值解和误差评估的功能。VEMcomp 还提供了大量示例来说明该库的使用方法。
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引用次数: 0
QMGI algorithm for solving quaternion equation and its application in color image encryption 求解四元数方程的 QMGI 算法及其在彩色图像加密中的应用
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-08-29 DOI: 10.1007/s11075-024-01920-x
Xinying Li, Caiqin Song, Hongjun Liu

In this present work, in order to solve the numerical solution of quaternion matrix equation (EY=F), the quaternion modified gradient-based algorithm (QMGI) is proposed by applying the real presentation of quaternion matrix. The proposed method can be applied to solve the quaternion solution, pure imaginary solution, and real solution of the studied equation (EY=F). If the studied equation is consistent, it is proved that the proposed algorithm converges to the exact solution for given any initial quaternion matrix under appropriate conditions. If the studied equation is not consistent, it is found that the QMGI algorithm converges to the least squares solution. And some numerical examples are examined to confirm the feasibility and efficiency of the proposed algorithms, which all indicate that the proposed QMGI algorithm is much more effective than QGI algorithm and QRGI algorithm in computational time and accuracy. Moreover, QMGl algorithm is applied to color image encryption and evaluated the encryption effectiveness from four aspects. All metrics are close to the ideal values. lt is demonstrated that the effectiveness of the encryption scheme and the accuracy of the obtained theory results in this paper.

在本研究中,为了求解四元矩阵方程(EY=F )的数值解,通过应用四元矩阵的实数表示,提出了基于梯度的四元修正算法(QMGI)。所提出的方法可用于求解所研究方程 (EY=F) 的四元数解、纯虚解和实数解。如果所研究的方程是一致的,那么在适当的条件下,对于给定的任意初始四元数矩阵,所提出的算法都能收敛到精确解。如果所研究的方程不一致,则会发现 QMGI 算法会收敛到最小二乘法解。为了证实所提算法的可行性和高效性,还通过一些数值实例进行了检验,结果表明所提 QMGI 算法在计算时间和计算精度上都远远优于 QGI 算法和 QRGI 算法。此外,还将 QMGl 算法应用于彩色图像加密,并从四个方面评估了加密效果。所有指标均接近理想值,证明了本文加密方案的有效性和理论结果的准确性。
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引用次数: 0
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Numerical Algorithms
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