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A fast continuous time approach with time scaling for nonsmooth convex optimization. 非光滑凸优化的快速连续时间方法。
Pub Date : 2022-01-01 DOI: 10.1186/s13662-022-03744-2
Radu Ioan Boţ, Mikhail A Karapetyants

In a Hilbert setting, we study the convergence properties of the second order in time dynamical system combining viscous and Hessian-driven damping with time scaling in relation to the minimization of a nonsmooth and convex function. The system is formulated in terms of the gradient of the Moreau envelope of the objective function with a time-dependent parameter. We show fast convergence rates for the Moreau envelope, its gradient along the trajectory, and also for the system velocity. From here, we derive fast convergence rates for the objective function along a path which is the image of the trajectory of the system through the proximal operator of the first. Moreover, we prove the weak convergence of the trajectory of the system to a global minimizer of the objective function. Finally, we provide multiple numerical examples illustrating the theoretical results.

在Hilbert条件下,研究了具有时间标度的二阶粘性和hessian驱动阻尼时间动力系统与非光滑凸函数最小化的收敛性。该系统是用目标函数的莫罗包络的梯度表示的,其参数与时间有关。我们展示了莫罗包络的快速收敛速率,它沿轨迹的梯度,以及系统速度。从这里,我们推导出目标函数沿路径的快速收敛速率,该路径是系统轨迹的图像,通过第一个算子的近端算子。此外,我们还证明了系统轨迹对目标函数的全局最小值的弱收敛性。最后,我们提供了多个数值例子来说明理论结果。
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引用次数: 4
A new mathematical model of multi-faced COVID-19 formulated by fractional derivative chains. 利用分数导数链建立的多面 COVID-19 新数学模型。
Pub Date : 2022-01-01 Epub Date: 2022-01-21 DOI: 10.1186/s13662-022-03677-w
Ibtisam Aldawish, Rabha W Ibrahim

It has been reported that there are seven different types of coronaviruses realized by individuals, containing those responsible for the SARS, MERS, and COVID-19 epidemics. Nowadays, numerous designs of COVID-19 are investigated using different operators of fractional calculus. Most of these mathematical models describe only one type of COVID-19 (infected and asymptomatic). In this study, we aim to present an altered growth of two or more types of COVID-19. Our technique is based on the ABC-fractional derivative operator. We investigate a system of coupled differential equations, which contains the dynamics of the diffusion between infected and asymptomatic people. The consequence is accordingly connected with a macroscopic rule for the individuals. In this analysis, we utilize the concept of a fractional chain. This type of chain is a fractional differential-difference equation combining continuous and discrete variables. The existence of solutions is recognized by formulating a matrix theory. The solution of the approximated system is shown to have a minimax point at the origin.

据报道,目前有七种不同类型的冠状病毒由个体实现,其中包括导致 SARS、MERS 和 COVID-19 流行的冠状病毒。目前,人们使用不同的分数微积分算子对 COVID-19 的多种设计进行了研究。这些数学模型大多只描述一种 COVID-19(感染和无症状)。在本研究中,我们旨在介绍两种或多种类型 COVID-19 的变化生长情况。我们的技术基于 ABC 分数导数算子。我们研究了一个耦合微分方程系统,其中包含感染者和无症状者之间的扩散动态。其结果相应地与个体的宏观规则相关联。在分析中,我们使用了分数链的概念。这种链是一种结合了连续变量和离散变量的分数微分差分方程。通过矩阵理论,我们认识到了解的存在。近似系统的解表明在原点有一个最小点。
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引用次数: 0
Effects of greenhouse gases and hypoxia on the population of aquatic species: a fractional mathematical model. 温室气体和缺氧对水生物种数量的影响:分数数学模型。
IF 2.3 Q1 MATHEMATICS Pub Date : 2022-01-01 Epub Date: 2022-04-15 DOI: 10.1186/s13662-022-03679-8
Pushpendra Kumar, V Govindaraj, Vedat Suat Erturk, Mohamed S Mohamed

Study of ecosystems has always been an interesting topic in the view of real-world dynamics. In this paper, we propose a fractional-order nonlinear mathematical model to describe the prelude of deteriorating quality of water cause of greenhouse gases on the population of aquatic animals. In the proposed system, we recall that greenhouse gases raise the temperature of water, and because of this reason, the dissolved oxygen level goes down, and also the rate of circulation of disintegrated oxygen by the aquatic animals rises, which causes a decrement in the density of aquatic species. We use a generalized form of the Caputo fractional derivative to describe the dynamics of the proposed problem. We also investigate equilibrium points of the given fractional-order model and discuss the asymptotic stability of the equilibria of the proposed autonomous model. We recall some important results to prove the existence of a unique solution of the model. For finding the numerical solution of the established fractional-order system, we apply a generalized predictor-corrector technique in the sense of proposed derivative and also justify the stability of the method. To express the novelty of the simulated results, we perform a number of graphs at various fractional-order cases. The given study is fully novel and useful for understanding the proposed real-world phenomena.

从现实世界的动力学角度来看,生态系统研究一直是一个有趣的课题。在本文中,我们提出了一个分数阶非线性数学模型,来描述温室气体导致的水质恶化对水生动物种群的影响。在该模型中,温室气体使水温升高,溶解氧水平下降,水生动物分解氧气的循环速率上升,从而导致水生物种密度下降。我们使用卡普托分数导数的广义形式来描述所提问题的动态。我们还研究了给定分数阶模型的平衡点,并讨论了所提自主模型平衡点的渐近稳定性。我们回顾了一些重要结果,以证明模型唯一解的存在。为了找到所建立的分数阶系统的数值解,我们应用了拟导数意义上的广义预测器-校正器技术,并证明了该方法的稳定性。为了表达模拟结果的新颖性,我们在各种分数阶情况下绘制了大量图形。该研究完全新颖,有助于理解所提出的现实世界现象。
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引用次数: 0
On three-term conjugate gradient method for optimization problems with applications on COVID-19 model and robotic motion control. 三项共轭梯度法在COVID-19模型和机器人运动控制中的应用
Pub Date : 2022-01-01 DOI: 10.1186/s13662-021-03638-9
Ibrahim Mohammed Sulaiman, Maulana Malik, Aliyu Muhammed Awwal, Poom Kumam, Mustafa Mamat, Shadi Al-Ahmad

The three-term conjugate gradient (CG) algorithms are among the efficient variants of CG algorithms for solving optimization models. This is due to their simplicity and low memory requirements. On the other hand, the regression model is one of the statistical relationship models whose solution is obtained using one of the least square methods including the CG-like method. In this paper, we present a modification of a three-term conjugate gradient method for unconstrained optimization models and further establish the global convergence under inexact line search. The proposed method was extended to formulate a regression model for the novel coronavirus (COVID-19). The study considers the globally infected cases from January to October 2020 in parameterizing the model. Preliminary results have shown that the proposed method is promising and produces efficient regression model for COVID-19 pandemic. Also, the method was extended to solve a motion control problem involving a two-joint planar robot.

三项共轭梯度(CG)算法是求解优化模型的一种有效的CG算法。这是由于它们的简单性和低内存需求。另一方面,回归模型是一种统计关系模型,其解是使用最小二乘法之一,包括类cg方法。本文对无约束优化模型的一种三项共轭梯度法进行了改进,进一步建立了非精确直线搜索下的全局收敛性。将该方法扩展到新型冠状病毒(COVID-19)的回归模型。该研究考虑了2020年1月至10月全球感染病例的参数化模型。初步结果表明,该方法具有较好的应用前景,能够建立有效的COVID-19大流行回归模型。并将该方法推广到两关节平面机器人的运动控制问题。
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引用次数: 17
Uniform convergence guarantees for the deep Ritz method for nonlinear problems. 非线性问题的深里兹方法的一致收敛保证。
Pub Date : 2022-01-01 Epub Date: 2022-07-15 DOI: 10.1186/s13662-022-03722-8
Patrick Dondl, Johannes Müller, Marius Zeinhofer

We provide convergence guarantees for the Deep Ritz Method for abstract variational energies. Our results cover nonlinear variational problems such as the p-Laplace equation or the Modica-Mortola energy with essential or natural boundary conditions. Under additional assumptions, we show that the convergence is uniform across bounded families of right-hand sides.

我们提供了抽象变分能量的深里兹方法的收敛性保证。我们的结果涵盖了非线性变分问题,如p-拉普拉斯方程或具有本质或自然边界条件的Modica-Mortola能量。在附加的假设下,我们证明了收敛性在右边有界族上是一致的。
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引用次数: 7
A delayed plant disease model with Caputo fractional derivatives. 带有卡普托分数导数的延迟植物病害模型。
IF 2.3 Q1 MATHEMATICS Pub Date : 2022-01-01 Epub Date: 2022-01-29 DOI: 10.1186/s13662-022-03684-x
Pushpendra Kumar, Dumitru Baleanu, Vedat Suat Erturk, Mustafa Inc, V Govindaraj

We analyze a time-delay Caputo-type fractional mathematical model containing the infection rate of Beddington-DeAngelis functional response to study the structure of a vector-borne plant epidemic. We prove the unique global solution existence for the given delay mathematical model by using fixed point results. We use the Adams-Bashforth-Moulton P-C algorithm for solving the given dynamical model. We give a number of graphical interpretations of the proposed solution. A number of novel results are demonstrated from the given practical and theoretical observations. By using 3-D plots we observe the variations in the flatness of our plots when the fractional order varies. The role of time delay on the proposed plant disease dynamics and the effects of infection rate in the population of susceptible and infectious classes are investigated. The main motivation of this research study is examining the dynamics of the vector-borne epidemic in the sense of fractional derivatives under memory effects. This study is an example of how the fractional derivatives are useful in plant epidemiology. The application of Caputo derivative with equal dimensionality includes the memory in the model, which is the main novelty of this study.

我们分析了一个包含贝丁顿-德安吉利斯函数反应感染率的时延卡普托型分数数学模型,以研究媒介传播植物流行病的结构。我们利用定点结果证明了给定延迟数学模型的唯一全局解存在性。我们使用 Adams-Bashforth-Moulton P-C 算法求解给定的动力学模型。我们对提出的解给出了一些图形解释。从给定的实践和理论观察中,我们展示了一些新的结果。通过使用三维图,我们观察到当分数阶数变化时,我们的图的平整度也会发生变化。我们还研究了时间延迟对所提出的植物病害动力学的作用,以及感染率对易感人群和感染人群的影响。这项研究的主要动机是考察记忆效应下分数导数意义上的病媒传染病动力学。本研究是分式导数在植物流行病学中的应用实例。等维度卡普托导数的应用包括模型中的记忆,这是本研究的主要创新之处。
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引用次数: 0
The inhomogeneous p-Laplacian equation with Neumann boundary conditions in the limit p→∞documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} 极限p中具有Neumann边界条件的非齐次p-拉普拉斯方程→∞documentclass[12pt]{minimal} usepackage{amsmath} use package{{wasysym}usepackage{amsfonts} usepackage{amssymb} userpackage{amsbsy}usepackage{mathrsfs} user package{upgeek}setlength{doddsidemargin}{-69pt}
Pub Date : 2021-12-14 DOI: 10.1186/s13662-023-03754-8
Leon Bungert
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引用次数: 1
A second-order low-regularity integrator for the nonlinear Schrödinger equation 非线性Schrödinger方程的二阶低正则积分器
Pub Date : 2021-09-02 DOI: 10.1186/s13662-022-03695-8
A. Ostermann, Yifei Wu, Fangyan Yao
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引用次数: 8
Path classification by stochastic linear recurrent neural networks 随机线性递归神经网络的路径分类
Pub Date : 2021-08-06 DOI: 10.1186/s13662-022-03686-9
Y. Boutaib, Wiebke Bartolomaeus, Sandra Nestler, H. Rauhut
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引用次数: 1
Operator compression with deep neural networks 算子压缩与深度神经网络
Pub Date : 2021-05-25 DOI: 10.1186/s13662-022-03702-y
Fabian Kröpfl, R. Maier, D. Peterseim
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引用次数: 9
期刊
Advances in continuous and discrete models
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