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Efficient estimates for matrix-inverse quadratic forms 矩阵反二次型的有效估计值
IF 2.2 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2025-02-01 DOI: 10.1016/j.apnum.2024.01.013
Emmanouil Bizas , Marilena Mitrouli , Ondřej Turek
In this paper we present two approaches for estimating matrix-inverse quadratic forms xTA1x, where A is a symmetric positive definite matrix of order n, and xRn. Using the first, analytic approach, we establish two families of estimates which are convenient for matrices with small condition number. Based on the second, heuristic approach, we derive two families of estimates which are suitable for matrices when vector x is close enough to an eigenvector. The low complexity and stability of the estimates is proved. Several numerical results illustrating the effectiveness of the methods are presented.
本文提出了估计矩阵逆二次型 xTA-1x 的两种方法,其中 A 是阶数为 n 的对称正定矩阵,x∈Rn。利用第一种分析方法,我们建立了两个估计族,这对条件数较小的矩阵很方便。基于第二种启发式方法,我们得出了两个估计族,当向量 x 与特征向量足够接近时,这两个估计族适用于矩阵。我们证明了这些估计值的低复杂性和稳定性。我们还给出了一些数值结果,以说明这些方法的有效性。
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引用次数: 0
A new fast algorithm for computing the mock-Chebyshev nodes 计算模拟切比雪夫节点的新型快速算法
IF 2.2 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2025-02-01 DOI: 10.1016/j.apnum.2024.03.002
B. Ali Ibrahimoglu
Interpolation by polynomials on equispaced points is not always convergent due to the Runge phenomenon, and also, the interpolation process is exponentially ill-conditioned. By taking advantage of the optimality of the interpolation processes on the Chebyshev-Lobatto nodes, one of the best strategies to defeat the Runge phenomenon is to use the mock-Chebyshev nodes for polynomial interpolation. Mock-Chebyshev nodes asymptotically follow the Chebyshev distribution, and they are selected from a sufficiently large set of equispaced nodes. However, there are few studies in the literature regarding the computation of these points.
In a recent paper [1], we have introduced a fast algorithm for computing the mock-Chebyshev nodes for a given set of (n+1) Chebyshev-Lobatto points using the distance between each pair of consecutive points. In this study, we propose a modification of the algorithm by changing the function to compute the quotient of the distance and show that this modified algorithm is also fast and stable; and gives a more accurate grid satisfying the conditions of a mock-Chebyshev grid with the complexity being O(n). Some numerical experiments using the points obtained by this modified algorithm are given to show its effectiveness and numerical results are also provided. A bivariate generalization of the mock-Chebyshev nodes to the Padua interpolation points is discussed.
由于 Runge 现象,等距点上的多项式插值并不总是收敛的,而且插值过程是指数条件不良的。利用切比雪夫-洛巴托节点上插值过程的最优性,克服 Runge 现象的最佳策略之一是使用模拟切比雪夫节点进行多项式插值。模拟切比雪夫节点在渐近上遵循切比雪夫分布,而且是从足够大的等距节点集中选择的。然而,文献中有关这些节点计算的研究很少。
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引用次数: 0
Special Volume on Numerical Analysis and Scientific Computation with Applications
IF 2.2 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2025-02-01 DOI: 10.1016/j.apnum.2024.10.019
Dr Khalide Jbilou
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引用次数: 0
Non trivial solutions for a system of coupled Ginzburg-Landau equations
IF 2.2 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2025-02-01 DOI: 10.1016/j.apnum.2024.10.010
Mariano De Leo , Juan Pablo Borgna , Cristian Huenchul
This article addresses both the existence and properties of non-trivial solutions for a system of coupled Ginzburg-Landau equations derived from nematic-superconducting models. Its main goal is to provide a thorough numerical description of the region in the parameter space containing solutions that behave as a mixed (non trivial) nematic-superconducting state along with a rigorous proof for the existence of this region. More precisely, the rigorous approach establishes that the parameter space is divided into two regions with qualitatively different properties, according to the magnitude of the coupling constant: for small values (weak coupling), there is a unique non-trivial solution, and for large values (strong coupling), only trivial solutions exist. In addition, using a shooting method-based numerical approach, the profiles for the nematic and superconducting components of the non trivial solution are given, together with an algorithm computing the transition values representing the boundaries for the weak coupling region: from superconducting to mixed, and from mixed to nematic. Finally, numerical evidence is given for the existence of a third region, related to neither a small nor a strong coupling parameter (medium coupling) for which multiple non trivial solutions exist.
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引用次数: 0
A mathematical model for studying the Red Blood Cell magnetic susceptibility 研究红血球磁感应强度的数学模型
IF 2.2 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2025-02-01 DOI: 10.1016/j.apnum.2024.05.014
Protopapas Eleftherios , Vafeas Panayiotis , Hadjinicolaou Maria
The susceptibility of the human Red Blood Cells (RBCs) under the action of magnetic fields, either serves as a biomarker in medical tests, e.g.. Magnetic Resonance Imaging, Nuclear Magnetic Resonance, Magnetoencephalography, or it is used in diagnostic and therapeutical processes, e.g.. magnetophoresis for cell sorting. In the present manuscript we provide analytical expressions for the magnetic potential and the magnetic field strength vector, when a magnetic field is applied to a RBC, modeled as a two-layered inverted spheroid. We introduce this way in the model the biconcave shape of the RBC and its structure (membrane and cytocol) in a more realistic representation, as until now, the RBC's shape was considered either as a sphere or a spheroid. The solution inside the RBC is obtained in R-separable form in terms of Legendre functions of the first and of the second kind and cyclic trigonometric functions, by applying appropriate boundary conditions on each layer. Our results reveal a non-uniform magnetic field inside the RBC. Parametric study of the solution, for various values of the physical properties of the RBC, is also provided, demonstrating the diamagnetic or the paramagnetic property of the RBC, which is strongly related to the health condition of the blood. The obtained solution may also serve for the justification of experimental results.
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引用次数: 0
The residual balanced IMEX decomposition for singly-diagonally-implicit schemes
IF 2.2 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2025-02-01 DOI: 10.1016/j.apnum.2024.09.030
Savio B. Rodrigues , Giovanni Belloni Fernandes Braga , Marcello Augusto Faraco de Medeiros
In numerical time-integration with implicit-explicit (IMEX) methods, a within-step adaptable decomposition called residual balanced (RB) decomposition is introduced. This new decomposition maintains time-stepping accuracy even when the implicit equation is only roughly approximated. This novel property is possible because a suitable modification of the traditional IMEX algorithm allows the remaining residual to be seamlessly transferred to the explicit part of the decomposition. The RB decomposition allows an early termination of iterations while preserving time-step accuracy. It can gain computational efficiency by exploring the trade-off between the computational effort placed in the iterative solver and the numerically stable step size. We develop a rigorous theory showing that RB maintains the order of singly diagonally implicit schemes. In computational experiments we show that, in many cases, RB-IMEX reduces the number of iterations when compared with the traditional IMEX method. It is often more stable also. The stability of RB-IMEX is studied using a model containing diffusion and dispersion; in this way, one can visualize how the stability region changes as a function of the number of iterations. Here, computational experiments use ESDIRK schemes for a stiff reaction-advection-diffusion equation, for a Navier-Stokes simulation with acoustic stiffness, and for a semi-implicit implementation of Burguers equation.
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引用次数: 0
On the growth factor of Hadamard matrices of order 20 论 20 阶哈达玛矩阵的增长因子
IF 2.2 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2025-02-01 DOI: 10.1016/j.apnum.2024.01.019
Emmanouil Lardas, Marilena Mitrouli
The aim of this work is to provide a complete list of all the possible values that the first six pivots of an Hadamard matrix of order 20 can take. This is accomplished by determining the possible values of certain minors of such matrices, in combination with the fact that the pivots can be computed in terms of these minors. We extend known results, by giving a different proof for the complete list of possible values for each of the first five pivots, as well as determining the complete list of possible values for the sixth pivot. By using a computational approach to search for new pivot patterns, we have also found at least 1246 different pivot patterns of Hadamard matrices of order 20.
这项工作的目的是提供一个完整的列表,列出阶数为 20 的哈达玛矩阵的前六个枢轴可能取的所有值。这是通过确定此类矩阵的某些最小值的可能值,并结合枢轴可以用这些最小值计算这一事实来实现的。我们扩展了已知结果,给出了前五个枢轴可能值的完整列表的不同证明,并确定了第六个枢轴可能值的完整列表。通过使用计算方法寻找新的枢轴模式,我们还发现了至少 1246 种不同的哈达玛矩阵 20 阶枢轴模式。
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引用次数: 0
The weak Galerkin finite element method for Stokes interface problems with curved interface
IF 2.2 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2025-02-01 DOI: 10.1016/j.apnum.2024.10.004
Lin Yang, Qilong Zhai, Ran Zhang
In this paper, we develop a weak Galerkin (WG) finite element scheme for the Stokes interface problems with curved interface. The conventional numerical schemes rely on the use of straight segments to approximate the curved interface and the accuracy is limited by geometric errors. Hence in our method, we directly construct the weak Galerkin finite element space on the curved cells to avoid geometric errors. For the integral calculation on curved cells, we employ non-affine transformations to map curved cells onto the reference element. The optimal error estimates are obtained in both the energy norm and the L2 norm. A series of numerical experiments are provided to validate the efficiency of the proposed WG method.
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引用次数: 0
An augmented Lagrangian approach for cardinality constrained minimization applied to variable selection problems 应用于变量选择问题的心数受限最小化的增强拉格朗日方法
IF 2.2 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2025-02-01 DOI: 10.1016/j.apnum.2023.12.006
N. Krejić , E.H.M. Krulikovski , M. Raydan
To solve convex constrained minimization problems, that also include a cardinality constraint, we propose an augmented Lagrangian scheme combined with alternating projection ideas. Optimization problems that involve a cardinality constraint are NP-hard mathematical programs and typically very hard to solve approximately. Our approach takes advantage of a recently developed and analyzed continuous formulation that relaxes the cardinality constraint. Based on that formulation, we solve a sequence of smooth convex constrained minimization problems, for which we use projected gradient-type methods. In our setting, the convex constraint region can be written as the intersection of a finite collection of convex sets that are easy and inexpensive to project. We apply our approach to a variety of over and under determined constrained linear least-squares problems, with both synthetic and real data that arise in variable selection, and demonstrate its effectiveness.
为了解决凸约束最小化问题(其中也包括万有引力约束),我们提出了一种结合交替投影思想的增强拉格朗日方案。涉及万有引力约束的优化问题是 NP 难数学程序,通常很难近似求解。我们的方法利用了最近开发和分析的连续公式,该公式放宽了万有引力约束。在此基础上,我们使用投影梯度法解决了一系列平滑的凸约束最小化问题。在我们的设置中,凸约束区域可以写成凸集的有限集合的交集,这些凸集易于投影且成本低廉。我们将我们的方法应用于各种过确定和欠确定的线性最小二乘约束问题,包括变量选择中出现的合成数据和真实数据,并证明了它的有效性。
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引用次数: 0
On the efficacy of conditioned and progressive Latin hypercube sampling in supervised machine learning 论监督式机器学习中条件式和渐进式拉丁超立方采样的功效
IF 2.2 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2025-02-01 DOI: 10.1016/j.apnum.2023.12.016
Ioannis Iordanis, Christos Koukouvinos, Iliana Silou
In this paper, Latin Hypercube Sampling (LHS) method is compared as per its effectiveness in supervised machine learning procedures. Employing LHS saves computer's processing time and in conjunction with Latin hypercube design properties and space filling ability, is considered as one of the most advanced mechanisms in terms of sampling. Although more data usually deliver better results, when using LHS techniques, same quality outputs can be produced with less data and, as a result, storage cost and training time are reduced. Conditioned Latin Hypercube Sampling (cLHS) is one of those techniques, successfully performing in supervised machine learning tasks. Unfortunately, the minimum sufficient training dataset size cannot be known in advance. In this case, progressive sampling is recommended since it begins with a small sample and progressively increases its size until model accuracy no longer improves. Combining Latin hypercube sampling and the idea of sequentially incrementing sampling, we test Progressive Latin Hypercube Sampling (PLHS) while monitoring the performance of the sampling-based training as the sample size grows. PLHS and cLHS algorithms are applied in datasets with discrete variables securing that each sample is provided with the Latin hypercube design properties and preserves the principal ability of LHS for space filling, as illustrated in respective sample projecting diagrams. The performance of the above LHS methods in supervised machine learning is evaluated by the degree of training of the model, which is certified through the accuracy of the produced confusion matrices in test files. The results from the use of the above Latin Hypercube Sampling techniques compared against benchmark sampling method empirically prove that machine learning training process becomes less costfull, while remaining reliable.
本文比较了拉丁超立方采样(LHS)方法在监督机器学习程序中的有效性。使用 LHS 可以节省计算机的处理时间,而且结合拉丁超立方设计特性和空间填充能力,LHS 被认为是最先进的采样机制之一。虽然更多的数据通常能带来更好的结果,但使用 LHS 技术时,用较少的数据也能产生相同质量的结果,从而降低了存储成本和培训时间。有条件拉丁超立方采样(cLHS)就是这样一种技术,它在监督机器学习任务中取得了成功。遗憾的是,训练数据集的最小足够大小无法预先知道。在这种情况下,建议采用渐进式采样,因为它从少量样本开始,逐步增加样本量,直到模型准确性不再提高为止。结合拉丁超立方采样和依次递增采样的思想,我们测试了渐进式拉丁超立方采样(PLHS),同时随着样本量的增加,监测基于采样训练的性能。PLHS 和 cLHS 算法适用于具有离散变量的数据集,确保每个样本都具有拉丁超立方设计特性,并保留了 LHS 空间填充的主要能力,如各自的样本投影图所示。上述 LHS 方法在有监督机器学习中的性能是通过模型的训练程度来评估的,而模型的训练程度则通过测试文件中产生的混淆矩阵的准确性来证明。上述拉丁超立方采样技术与基准采样方法的比较结果证明,机器学习训练过程的成本更低,同时保持了可靠性。
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Applied Numerical Mathematics
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