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Variable parameter Uzawa method for solving the indefinite least squares problem 求解不定最小二乘法问题的可变参数乌泽法
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-31 DOI: 10.1007/s11075-024-01905-w
Lingsheng Meng, Kailiang Xin, Jun Li

In this paper, the variable parameter Uzawa method is presented to solve the indefinite least squares problem. The proposed iterative method is unconditionally convergent, and its iterative algorithm and parameter designing are simple and efficient. Numerical experiments show that the variable parameter Uzawa method is superior to the USSOR method (Song, Int. J. Comput. Math. 97, 1781–1791 2020) and the splitting-based randomized iterative method (Zhang and Li, Appl. Math. Comput. 446, 127892 2023).

本文提出了解决不定最小二乘法问题的变参数乌泽法。本文提出的迭代法无条件收敛,其迭代算法和参数设计简单高效。数值实验表明,变参数 Uzawa 方法优于 USSOR 方法(Song,Int. J. Comput. Math. 97,1781-1791 2020)和基于分裂的随机迭代法(Zhang 和 Li,Appl.)
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引用次数: 0
Refined and refined harmonic Jacobi–Davidson methods for computing several GSVD components of a large regular matrix pair 计算大型正则矩阵对的多个 GSVD 分量的改进和改进谐波雅各比-戴维森方法
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-31 DOI: 10.1007/s11075-024-01901-0
Jinzhi Huang, Zhongxiao Jia

Three refined and refined harmonic extraction-based Jacobi–Davidson (JD) type methods are proposed, and their thick-restart algorithms with deflation and purgation are developed to compute several generalized singular value decomposition (GSVD) components of a large regular matrix pair. The new methods are called refined cross product-free (RCPF), refined cross product-free harmonic (RCPF-harmonic) and refined inverse-free harmonic (RIF-harmonic) JDGSVD algorithms, abbreviated as RCPF-JDGSVD, RCPF-HJDGSVD and RIF-HJDGSVD, respectively. The new JDGSVD methods are more efficient than the corresponding standard and harmonic extraction-based JDSVD methods proposed previously by the authors, and can overcome the erratic behavior and intrinsic possible non-convergence of the latter ones. Numerical experiments illustrate that RCPF-JDGSVD performs better for the computation of extreme GSVD components while RCPF-HJDGSVD and RIF-HJDGSVD are more suitable for that of interior GSVD components.

本文提出了三种基于雅各比-戴维森(JD)类型的精炼谐波提取方法,并开发了其具有放缩和净化功能的厚起算法,用于计算大型正则矩阵对的若干广义奇异值分解(GSVD)分量。新方法被称为精炼无交叉积(RCPF)、精炼无交叉积谐波(RCPF-谐波)和精炼无逆谐波(RIF-谐波)JDGSVD 算法,分别简称为 RCPF-JDGSVD、RCPF-HJDGSVD 和 RIF-HJDGSVD。新的 JDGSVD 方法比作者之前提出的相应标准 JDSVD 方法和基于谐波提取的 JDSVD 方法更有效,并能克服后者的不稳定行为和内在可能的不收敛性。数值实验表明,RCPF-JDGSVD 在计算 GSVD 极值分量时表现更好,而 RCPF-HJDGSVD 和 RIF-HJDGSVD 更适合计算 GSVD 内部分量。
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引用次数: 0
A Tikhonov-type regularization method for Caputo fractional derivative 卡普托分数导数的提霍诺夫式正则化方法
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-30 DOI: 10.1007/s11075-024-01883-z
Nguyen Van Duc, Thi-Phong Nguyen, Nguyen Phuong Ha, Nguyen The Anh, Luu Duc Manh, Hoang Cong Gia Bao

Stability estimates of Hölder type for the problem of evaluating the Caputo fractional derivative are obtained. This ill-posed problem is regularized by a Tikhonov-type method, which guarantees error estimates of Hölder type. Numerical results are presented to confirm the theory.

针对卡普托分数导数的求值问题,获得了霍尔德类型的稳定性估计。这个问题是由一种 Tikhonov 型方法正则化的,它保证了霍尔德类型的误差估计。数值结果证实了这一理论。
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引用次数: 0
Generalized weak Galerkin finite element method for linear elasticity interface problems 线性弹性界面问题的广义弱 Galerkin 有限元方法
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-29 DOI: 10.1007/s11075-024-01904-x
Yue Wang, Fuzheng Gao

A generalized weak Galerkin finite element method for linear elasticity interface problems is presented. The generalized weak gradient (divergence) is consisted of classical gradient (divergence) and the solution of local problem. Thus, the finite element space can be extended to arbitrary combination of piecewise polynomial spaces. The error equation and error estimates are proved. The numerical results illustrate the efficiency and flexibility for different interfaces, partitions and combinations, the locking-free property, the well performance for low regularity solution in discrete energy, (L^2) and (L^{infty }) norms. Meanwhile, we present the numerical comparison between our algorithm and the weak Galerkin finite element algorithm to demonstrate the flexibility of our algorithm. In addition, for some cases, the convergence rates in numerical tests are obviously higher than the theoretical prediction for the smooth and low regularity solutions.

介绍了线性弹性界面问题的广义弱 Galerkin 有限元方法。广义弱梯度(发散)由经典梯度(发散)和局部问题解组成。因此,有限元空间可以扩展到片断多项式空间的任意组合。证明了误差方程和误差估计。数值结果表明了不同界面、分区和组合的效率和灵活性、无锁定特性、离散能量、(L^2)和(L^{infty })规范下低正则性求解的良好性能。同时,我们给出了我们的算法与弱 Galerkin 有限元算法的数值比较,以证明我们算法的灵活性。此外,在某些情况下,对于平滑低正则解,数值试验的收敛速率明显高于理论预测。
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引用次数: 0
Extended explicit Pseudo two-step Runge-Kutta-Nyström methods for general second-order oscillatory systems 一般二阶振荡系统的扩展显式伪两步 Runge-Kutta-Nyström 方法
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-29 DOI: 10.1007/s11075-024-01896-8
Yonglei Fang, Changying Liu, Xiong You

Explicit pseudo two-step extended Runge-Kutta-Nyström (EPTSERKN) methods for the numerical integration of general second-order oscillatory differential systems are discussed in this paper. New explicit pseudo two-step Runge-Kutta-Nyström (EPTSRKN) methods and explicit extended Runge-Kutta-Nyström (ERKN) methods are derived. We give the global error analysis of the new methods. The s-stages new methods are of order (s+1) with some suitable nodes. Numerical experiments are carried out to show the efficiency and robustness of the new methods.

本文讨论了用于一般二阶振荡微分系统数值积分的显式伪两步扩展 Runge-Kutta-Nyström (EPTSERKN) 方法。推导了新的显式伪两步 Runge-Kutta-Nyström (EPTSRKN) 方法和显式扩展 Runge-Kutta-Nyström (ERKN) 方法。我们给出了新方法的全局误差分析。新方法的 s 阶为 (s+1) 阶,有一些合适的节点。我们通过数值实验证明了新方法的效率和鲁棒性。
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引用次数: 0
Fast and accurate numerical algorithm for solving stochastic Itô-Volterra integral equations 求解随机伊托-伏特拉积分方程的快速准确数值算法
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-29 DOI: 10.1007/s11075-024-01898-6
Rebiha Zeghdane

The purpose of this paper is to present a simple numerical technique for approximating the solutions of stochastic Volterra integral equations. The proposed method depends on the Picard iteration and uses a suitable quadrature rule. Error estimates and associated theorems have been proved for this proposed technique. Some test examples have been studied to verify the applicability and accuracy of the proposed technique.

本文旨在介绍一种用于逼近随机 Volterra 积分方程解的简单数值技术。所提出的方法依赖于 Picard 迭代,并使用合适的正交规则。本文证明了所提技术的误差估计和相关定理。研究了一些测试实例,以验证所提技术的适用性和准确性。
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引用次数: 0
Limited memory gradient methods for unconstrained optimization 无约束优化的有限记忆梯度法
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-26 DOI: 10.1007/s11075-024-01895-9
Giulia Ferrandi, Michiel E. Hochstenbach

The limited memory steepest descent method (LMSD, Fletcher, 2012) for unconstrained optimization problems stores a few past gradients to compute multiple stepsizes at once. We review this method and propose new variants. For strictly convex quadratic objective functions, we study the numerical behavior of different techniques to compute new stepsizes. In particular, we introduce a method to improve the use of harmonic Ritz values. We also show the existence of a secant condition associated with LMSD, where the approximating Hessian is projected onto a low-dimensional space. In the general nonlinear case, we propose two new alternatives to Fletcher’s method: first, the addition of symmetry constraints to the secant condition valid for the quadratic case; second, a perturbation of the last differences between consecutive gradients, to satisfy multiple secant equations simultaneously. We show that Fletcher’s method can also be interpreted from this viewpoint.

用于无约束优化问题的有限记忆最陡梯度下降法(LMSD,Fletcher,2012 年)会存储一些过去的梯度,以便一次计算多个步长。我们回顾了这种方法,并提出了新的变体。对于严格凸二次目标函数,我们研究了计算新步长的不同技术的数值行为。特别是,我们介绍了一种改进谐波里兹值使用的方法。我们还证明了与 LMSD 相关的secant 条件的存在,其中近似 Hessian 被投影到一个低维空间上。在一般非线性情况下,我们提出了弗莱彻方法的两个新替代方案:第一,在二次方程情况下有效的secant条件中添加对称约束;第二,对连续梯度之间的最后差值进行扰动,以同时满足多个secant方程。我们证明,弗莱彻方法也可以从这个角度进行解释。
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引用次数: 0
Computation of polynomial and rational approximations in complex domains by the $$tau $$ -method 用 $$tau$ 方法计算复域中的多项式和有理近似值
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-25 DOI: 10.1007/s11075-024-01897-7
Irina Georgieva, Clemens Hofreither

We investigate numerical methods for computation of polynomial and rational approximations of functions in complex domains based on Faber polynomials and the Lanczos (tau )-method. Our interest is motivated by applications in fractional partial differential equations. We give an overview of previous results related to the basis of Faber polynomials associated with a complex domain, Faber expansion, and the Lanczos (tau )-method. We also collect numerical algorithms for the computational realization of these concepts. Our main new contribution is a (tau )-method for rational approximation in complex domains which uses Faber polynomials in the perturbation term. We realize it via a novel hybrid symbolic-numeric algorithm which can be applied to arbitrary functions satisfying a suitable differential equation. We present some numerical examples, where we use sectors lying in the complex plane as our domains of interest. We compare results for the various polynomial and rational approximation techniques outlined above; in particular, we observe exponential convergence with respect to the rational degree for our proposed method.

我们研究了基于法布尔多项式和 Lanczos (tau )方法计算复域中函数的多项式和有理近似的数值方法。我们的兴趣源于分数偏微分方程中的应用。我们概述了与复数域相关的法布尔多项式基础、法布尔展开和 Lanczos (tau )方法有关的先前结果。我们还收集了计算实现这些概念的数值算法。我们的主要新贡献是在复杂域中使用法布尔多项式进行有理逼近的((tau )-method)扰动项。我们通过一种新颖的符号-数值混合算法来实现它,这种算法可以应用于满足适当微分方程的任意函数。我们介绍了一些数值示例,其中我们使用复平面内的扇形作为我们感兴趣的域。我们比较了上述各种多项式和有理近似技术的结果;特别是,我们观察到我们提出的方法在有理程度上呈指数收敛。
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引用次数: 0
Unified convergence analysis of a class of iterative methods 一类迭代法的统一收敛分析
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-24 DOI: 10.1007/s11075-024-01893-x
Muniyasamy M, Santhosh George, Chandhini G

In this paper, unified convergence analyses for a class of iterative methods of order three, five, and six are studied to solve the nonlinear systems in Banach space settings. Our analysis gives the number of iterations needed to achieve the given accuracy and the radius of the convergence ball precisely using weaker conditions on the involved operator. Various numerical examples have been taken to illustrate the proposed method, and the theoretical convergence has been validated via these examples.

本文研究了一类三阶、五阶和六阶迭代法的统一收敛分析,以求解巴拿赫空间环境下的非线性系统。我们的分析精确地给出了达到给定精度所需的迭代次数,以及利用相关算子的较弱条件得出的收敛球半径。我们通过各种数值示例来说明所提出的方法,并通过这些示例验证了理论上的收敛性。
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引用次数: 0
On solving a revised model of the nonnegative matrix factorization problem by the modified adaptive versions of the Dai–Liao method 关于用戴廖法的修正自适应版本求解非负矩阵因式分解问题的修正模型
IF 2.1 3区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-24 DOI: 10.1007/s11075-024-01886-w
Saman Babaie-Kafaki, Fatemeh Dargahi, Zohre Aminifard

We suggest a revised form of a classic measure function to be employed in the optimization model of the nonnegative matrix factorization problem. More exactly, using sparse matrix approximations, the revision term is embedded to the model for penalizing the ill-conditioning in the computational trajectory to obtain the factorization elements. Then, as an extension of the Euclidean norm, we employ the ellipsoid norm to gain adaptive formulas for the Dai–Liao parameter in a least-squares framework. In essence, the parametric choices here are obtained by pushing the Dai–Liao direction to the direction of a well-functioning three-term conjugate gradient algorithm. In our scheme, the well-known BFGS and DFP quasi–Newton updating formulas are used to characterize the positive definite matrix factor of the ellipsoid norm. To see at what level our model revisions as well as our algorithmic modifications are effective, we seek some numerical evidence by conducting classic computational tests and assessing the outputs as well. As reported, the results weigh enough value on our analytical efforts.

我们建议在非负矩阵因式分解问题的优化模型中采用经典度量函数的修正形式。更确切地说,利用稀疏矩阵近似,修正项被嵌入到模型中,以惩罚计算轨迹中的条件不良,从而获得因式分解元素。然后,作为欧氏规范的扩展,我们采用椭圆规范,在最小二乘框架下获得戴辽参数的自适应公式。实质上,这里的参数选择是通过将傣辽方向推向功能良好的三项共轭梯度算法的方向而获得的。在我们的方案中,著名的 BFGS 和 DFP 准牛顿更新公式被用来描述椭球体规范的正定矩阵因子。为了了解我们的模型修正和算法修改在多大程度上是有效的,我们通过进行经典的计算测试和评估输出结果来寻求一些数字证据。正如报告所述,这些结果足以证明我们的分析工作是有价值的。
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Numerical Algorithms
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