Applied Mathematics and Computation最新文献

英文中文

A step function based recursion method for 0/1 deep neural networks 基于阶跃函数的 0/1 深度神经网络递归方法

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-21 DOI: 10.1016/j.amc.2024.129129

Hui Zhang , Shenglong Zhou , Geoffrey Ye Li , Naihua Xiu , Yiju Wang

The deep neural network with step function activation (0/1 DNNs) is a fundamental composite model in deep learning which has high efficiency and robustness to outliers. However, due to the discontinuity and lacking subgradient information of the 0/1 DNNs model, prior researches are largely focused on designing continuous functions to approximate the step activation and developing continuous optimization methods. In this paper, by introducing two sets of network node variables into the 0/1 DNNs and by exploring the composite structure of the resulted model, the 0/1 DNNs is decomposed into a unary optimization model associated with the step function and three derivational optimization subproblems associated with the other variables. For the unary optimization model and two derivational optimization subproblems, we present a closed form solution, and for the third derivational optimization subproblem, we propose an efficient proximal method. Based on this, a globally convergent step function based recursion method for the 0/1 DNNs is developed. The efficiency and performance of the proposed algorithm are validated via theoretical analysis as well as some illustrative numerical examples on classifying MNIST, FashionMNIST and Cifar10 datasets.

带阶跃函数激活的深度神经网络（0/1 DNNs）是深度学习中的一种基本复合模型，具有高效率和对异常值的鲁棒性。然而，由于 0/1 DNNs 模型的不连续性和缺乏子梯度信息，之前的研究主要集中在设计近似阶跃激活的连续函数和开发连续优化方法上。本文通过在 0/1 DNNs 中引入两组网络节点变量，并通过探索所得模型的复合结构，将 0/1 DNNs 分解为与阶跃函数相关的一元优化模型和与其他变量相关的三个衍生优化子问题。对于一元优化模型和两个派生优化子问题，我们提出了闭式解法；对于第三个派生优化子问题，我们提出了一种高效的近似方法。在此基础上，我们为 0/1 DNN 开发了一种基于阶跃函数的全局收敛递归方法。通过理论分析以及在 MNIST、FashionMNIST 和 Cifar10 数据集分类上的一些数值示例，验证了所提算法的效率和性能。

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引用次数: 0

Linear and nonlinear filtering for a two-layer quasi-geostrophic ocean model 双层准地养海洋模型的线性和非线性滤波

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-21 DOI: 10.1016/j.amc.2024.129121

Lander Besabe , Michele Girfoglio , Annalisa Quaini , Gianluigi Rozza

Although the two-layer quasi-geostrophic equations (2QGE) are a simplified model for the dynamics of a stratified, wind-driven ocean, their numerical simulation is still plagued by the need for high resolution to capture the full spectrum of turbulent scales. Since such high resolution would lead to unreasonable computational times, it is typical to resort to coarse low-resolution meshes combined with the so-called eddy viscosity parameterization to account for the diffusion mechanisms that are not captured due to mesh under-resolution. We propose to enable the use of further coarsened meshes by adding a (linear or nonlinear) differential low-pass filter to the 2QGE, without changing the eddy viscosity coefficient. While the linear filter introduces constant (additional) artificial viscosity everywhere in the domain, the nonlinear filter relies on an indicator function to determine where and how much artificial viscosity is needed. Through several numerical results for a double-gyre wind forcing benchmark, we show that with the nonlinear filter we obtain accurate results with very coarse meshes, thereby drastically reducing the computational time (speed up ranging from 30 to 300).

尽管两层准地转方程（2QGE）是一个简化的分层风动海洋动力学模型，但其数值模拟仍然受到需要高分辨率以捕捉全部湍流尺度谱的困扰。由于这种高分辨率会导致不合理的计算时间，因此通常采用粗糙的低分辨率网格并结合所谓的涡粘参数化来解释由于网格分辨率不足而无法捕捉到的扩散机制。我们建议通过在 2QGE 中添加（线性或非线性）微分低通滤波器，在不改变涡粘系数的情况下使用进一步粗化的网格。线性滤波器在域内各处引入恒定（额外）的人工粘性，而非线性滤波器则依赖于一个指示函数来确定在何处以及需要多少人工粘性。通过几个双褶皱风力强迫基准的数值结果，我们表明，使用非线性滤波器，我们可以用非常粗的网格获得精确的结果，从而大大缩短计算时间（加速度从 30 到 300 不等）。

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引用次数: 0

Matrix expressions of symmetric n-player games 对称 n 人游戏的矩阵表达式

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-17 DOI: 10.1016/j.amc.2024.129134

Yuanhua Wang , Ying Wang , Haitao Li , Wenke Zang

The symmetric property in n-player games is explored in this paper. First of all, some algebraically verifiable conditions are provided respectively for the totally symmetric, weakly symmetric and anonymous games, which are based on the construction of permutation matrix, and some interesting properties are revealed. Then we investigate how network structures affect the symmetry in a networked game. Some sufficient conditions are proposed and several nice properties are preserved when traditional networked games are extended to generalized networked games, in which the resulting network structure is more complex than a classical network graph.

本文探讨了 n 人博弈中的对称属性。首先，本文基于排列矩阵的构造，分别为完全对称博弈、弱对称博弈和匿名博弈提供了一些可验证的代数条件，并揭示了一些有趣的性质。然后，我们研究了网络结构如何影响网络博弈的对称性。当传统网络博弈扩展到广义网络博弈时，我们提出了一些充分条件，并保留了一些很好的性质，在广义网络博弈中，所产生的网络结构比经典网络图更复杂。

引用次数: 0

Mutual-visibility and general position in double graphs and in Mycielskians 双图和迈锡尔图中的互见性和一般位置

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-17 DOI: 10.1016/j.amc.2024.129131

Dhanya Roy , Sandi Klavžar , Aparna Lakshmanan S

The general position problem in graphs is to find the largest possible set of vertices with the property that no three of them lie on a common shortest path. The mutual-visibility problem in graphs is to find the maximum number of vertices that can be selected such that every pair of vertices in the collection has a shortest path between them with no vertex from the collection as an internal vertex. Here, the general position problem and the mutual-visibility problem are investigated in double graphs and in Mycielskian graphs. Sharp general bounds are proved, in particular involving the total and the outer mutual-visibility number of base graphs. Several exact values are also determined, in particular the mutual-visibility number of the double graphs and of the Mycielskian of cycles.

图中的一般位置问题是找到最大可能的顶点集合，其性质是其中没有三个顶点位于共同的最短路径上。图中的互见性问题是找到可以选择的最大顶点数，使得集合中的每对顶点之间都有一条最短路径，且集合中没有顶点作为内部顶点。这里研究的是双图和迈锡尔图中的一般位置问题和互见性问题。证明了尖锐的一般界限，特别是涉及基图的总互见数和外互见数。此外，还确定了几个精确值，特别是双图和循环 Mycielskian 的互见数。

引用次数: 0

Improving approximation accuracy in Godunov-type smoothed particle hydrodynamics methods 提高戈杜诺夫型平滑粒子流体力学方法的近似精度

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-16 DOI: 10.1016/j.amc.2024.129128

G.D. Rublev , A.N. Parshikov , S.A. Dyachkov

The study examines the origin of errors resulting from the approximation of the right hand sides of the Euler equations using the Godunov type contact method of smoothed particle hydrodynamics (CSPH). The analytical expression for the numerical shear viscosity in CSPH method is obtained. In our recent study the numerical viscosity was determined by comparing the numerical solution of momentum diffusion in the shear flow with theoretical one. In this study we deduce the analytical expression for the numerical viscosity which is found to be similar to numerical one, confirming the obtained results. To reduce numerical diffusion, diffusion limiters are typically applied to expressions for contact values of velocity and pressure, as well as higher-order reconstruction schemes. Based on the performed theoretical analysis, we propose a new method for correcting quantities at interparticle contacts in CSPH method, which can be easily extended to the MUSCL-type (Monotonic Upstream-centered Scheme for Conservation Laws) method. Original CSPH and MUSCL-SPH approaches and ones with aforementioned correction are compared.

研究探讨了使用平滑粒子流体力学（CSPH）的戈杜诺夫接触法对欧拉方程右边进行近似所产生误差的根源。我们得到了 CSPH 方法中数值剪切粘度的解析表达式。在我们最近的研究中，通过比较剪切流中动量扩散的数值解与理论解，确定了数值粘度。在本研究中，我们推导出了数值粘度的分析表达式，发现其与数值粘度相似，从而证实了所获得的结果。为了减少数值扩散，通常会在速度和压力的接触值表达式以及高阶重构方案中应用扩散限制器。基于所做的理论分析，我们提出了一种在 CSPH 方法中修正粒子间接触量的新方法，该方法可以很容易地扩展到 MUSCL 类型（单调上游中心守恒定律方案）方法。比较了原始的 CSPH 和 MUSCL-SPH 方法以及采用上述修正的方法。

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引用次数: 0

Analysis and control of demand response in smart grids: An evolutionary game method 智能电网中需求响应的分析与控制：进化博弈法

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-16 DOI: 10.1016/j.amc.2024.129130

Mengyu Zhou , Xingwen Liu , Qi Hu , Feng Shu

As an effective strategy for load management in smart grids, demand response establishes a bidirectional connection between the electricity supplier and users. Based on the networked evolutionary game theory, this paper studies the demand-response issue for a class of smart grids by using the semi-tensor product of matrices. The paper proceeds as follows. (i) Considering the dynamic interactions between the supplier and users, the demand response is modeled as a heterogeneous networked evolutionary game and is expressed as dynamical form by semi-tensor product. (ii) A sufficient and necessary condition is provided to verify the convergence to a fixed point of the considered system. (iii) A feedback controller is designed to ensure the system electricity consumption and price to maintain at a desired level. Finally, an example is presented to illustrate the feasibility of the proposed method.

作为智能电网负荷管理的一种有效策略，需求响应在电力供应商和用户之间建立了一种双向联系。本文基于网络演化博弈论，利用矩阵的半张量乘法研究了一类智能电网的需求响应问题。本文的研究过程如下(i) 考虑到供应商和用户之间的动态互动，将需求响应建模为异构网络化演化博弈，并用半张量积表示为动态形式。(ii) 提供了一个充分和必要条件来验证所考虑的系统是否收敛到一个固定点。(iii) 设计了一个反馈控制器，以确保系统耗电量和电价维持在理想水平。最后，举例说明了所提方法的可行性。

引用次数: 0

Deep-time neural networks: An efficient approach for solving high-dimensional PDEs 深度时间神经网络：解决高维 PDE 的高效方法

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-16 DOI: 10.1016/j.amc.2024.129117

Ahmad Aghapour , Hamid Arian , Luis Seco

This paper presents the Deep-Time Neural Network (DTNN), an efficient and novel deep-learning approach for solving partial differential equations (PDEs). DTNN leverages the power of deep neural networks to approximate the solution for a class of quasi-linear parabolic PDEs. We demonstrate that DTNN significantly reduces the computational cost and speeds up the training process compared to other models in the literature. The results of our study indicate that DTNN architecture is promising for the fast and accurate solution of time-dependent PDEs in various scientific and engineering applications. The DTNN architecture addresses the pressing need for enhanced time considerations in the deeper layers of Artificial Neural Networks (ANNs), thereby improving convergence time for high-dimensional PDE solutions. This is achieved by integrating time into the hidden layers of the DTNN, demonstrating a marked improvement over existing ANN-based solutions regarding efficiency and speed.

本文介绍了用于求解偏微分方程（PDEs）的高效、新颖的深度学习方法--深度时间神经网络（DTNN）。DTNN 利用深度神经网络的强大功能来近似求解一类准线性抛物线偏微分方程。我们证明，与文献中的其他模型相比，DTNN 大大降低了计算成本，加快了训练过程。我们的研究结果表明，DTNN 架构有望在各种科学和工程应用中快速、准确地解决随时间变化的多项式方程。DTNN 架构解决了在人工神经网络（ANN）深层加强时间考虑的迫切需求，从而改善了高维 PDE 解法的收敛时间。这是通过将时间整合到 DTNN 的隐藏层来实现的，在效率和速度方面与现有的基于人工神经网络的解决方案相比有了明显的改进。

引用次数: 0

Insider trading at a random deadline with correlation between dynamic asset and stochastic liquidity 具有动态资产和随机流动性相关性的随机截止日期内幕交易

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-16 DOI: 10.1016/j.amc.2024.129120

Jixiu Qiu , Yonghui Zhou

We propose a generalized continuous-time insider trading model, building upon the frameworks of Caldentey and Stacchetti (2010) and Collin-Dufresne and Fos (2016), with a correlation between the value of a risky asset following an Ornstein-Uhlenbeck-type process and the noise trading volume with volatility characterized by a general stochastic process. And a closed form of the market equilibrium is established, consisting of the insider's trading strategy and the market makers' pricing rule. It shows that at the equilibrium: (i) all of the insider's private information is released at the end of the transaction; (ii) market depth and market liquidity evolve as semi-martingales, respectively; and (iii) the equilibrium price is driven by a bridge process that solves an Ornstein-Uhlenbeck-type SDE. Numerical simulations show that as the correlation coefficient increases, the equilibrium price becomes more informative, leading to a decrease in both the trading intensity and the expected payoff for the insider.

我们在 Caldentey 和 Stacchetti（2010 年）以及 Collin-Dufresne 和 Fos（2016 年）的框架基础上提出了一个广义连续时间内幕交易模型，该模型中风险资产的价值遵循 Ornstein-Uhlenbeck 型过程，而噪音交易量的波动性则以一般随机过程为特征。由内部人的交易策略和做市商的定价规则组成，建立了市场均衡的封闭形式。结果表明，在均衡状态下：(i) 内幕交易者的所有私人信息都会在交易结束时被释放；(ii) 市场深度和市场流动性分别以半鞅形式演化；(iii) 均衡价格由求解奥恩斯坦-乌伦贝克型 SDE 的桥过程驱动。数值模拟表明，随着相关系数的增加，均衡价格的信息量会越来越大，从而导致交易强度和内部人预期收益的下降。

引用次数: 0

A Lagrange barrier approach for the minimum concave cost supply problem via a logarithmic descent direction algorithm 通过对数下降方向算法解决最小凹成本供应问题的拉格朗日障碍法

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-16 DOI: 10.1016/j.amc.2024.129114

Yaolong Yu , Zhengtian Wu , Baoping Jiang , Huaicheng Yan , Yichen Lu

The minimisation of concave costs in the supply chain presents a challenging non-deterministic polynomial (NP) optimisation problem, widely applicable in industrial and management engineering. To approximate solutions to this problem, we propose a logarithmic descent direction algorithm (LDDA) that utilises the Lagrange logarithmic barrier function. As the barrier variable decreases from a high positive value to zero, the algorithm is capable of tracking the minimal track of the logarithmic barrier function, thereby obtaining top-quality solutions. The Lagrange function is utilised to handle linear equality constraints, whilst the logarithmic barrier function compels the solution towards the global or near-global optimum. Within this concave cost supply model, a logarithmic descent direction is constructed, and an iterative optimisation process for the algorithm is proposed. A corresponding Lyapunov function naturally emerges from this descent direction, thus ensuring convergence of the proposed algorithm. Numerical results demonstrate the effectiveness of the algorithm.

供应链中凹成本的最小化是一个具有挑战性的非确定多项式（NP）优化问题，广泛应用于工业和管理工程领域。为了近似解决这一问题，我们提出了一种利用拉格朗日对数障碍函数的对数下降方向算法（LDDA）。当障碍变量从高正值下降到零时，该算法能够跟踪对数障碍函数的最小轨迹，从而获得高质量的解决方案。拉格朗日函数用于处理线性相等约束条件，而对数障碍函数则迫使求解走向全局或接近全局最优。在这个凹成本供应模型中，构建了一个对数下降方向，并提出了算法的迭代优化过程。相应的 Lyapunov 函数会从这个下降方向自然产生，从而确保所提算法的收敛性。数值结果证明了该算法的有效性。

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引用次数: 0

Jacobi spectral collocation method of space-fractional Navier-Stokes equations 空间分数纳维-斯托克斯方程的雅可比谱配位法

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-15 DOI: 10.1016/j.amc.2024.129111

Yujian Jiao , Tingting Li , Zhongqiang Zhang

In this paper, we study the Jacobi spectral collocation method for two-dimensional space-fractional Navier-Stokes equations with Laplacian and fractional Laplacian. We first derive modified fractional differentiation matrices to accommodate the singularity in two dimensions and verify the boundedness of its spectral radius. Next, we construct a fully discrete scheme for the space-fractional Navier-Stokes equations, combined with the first-order implicit-explicit Euler time-stepping scheme at the Jacobi-Gauss-Lobatto collocation points. Through some two-dimensional numerical examples, we present the influence of different parameters in the equations on numerical errors. Various numerical examples verify the effectiveness of our method and suggest the smoothness of the solution for further regularity analysis.

本文研究了带有拉普拉奇和分数拉普拉奇的二维空间分数 Navier-Stokes 方程的雅可比谱配位法。我们首先推导出修正的分数微分矩阵，以适应二维的奇异性，并验证了其谱半径的有界性。接下来，我们为空间分数 Navier-Stokes 方程构建了一个完全离散的方案，并在 Jacobi-Gauss-Lobatto 配点上结合了一阶隐式-显式欧拉时间步进方案。通过一些二维数值示例，我们介绍了方程中不同参数对数值误差的影响。各种数值示例验证了我们方法的有效性，并为进一步的正则性分析提出了解的平滑性建议。

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Applied Mathematics and Computation

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