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Correction to: Optimal Error Estimates for Gegenbauer Approximations in Fractional Spaces

Journal of scientific computing 2025年第2期103卷 1-2页

作者： Xie, Ruiyi Liu, Wenjie Wang, Haiyong Wu, Boying School of Mathematics Harbin Institute of Technology Harbin People’s Republic of China Key Laboratory of Computing and Stochastic Mathematics Ministry of Education (LCSM) Changsha People’s Republic of China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan People’s Republic of China China and Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan People’s Republic of China

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Learning with Noisy Labels: the Exploration of Error Bounds in Classification

arXiv

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arXiv 2025年

作者： Liu, Haixia Li, Boxiao Yang, Can Wang, Yang School of Mathematics and Statistics Institute of Interdisciplinary Research for Mathematics and Applied Science Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan China School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Wuhan China Department of Mathematics The Hong Kong University of Science and Technology Clear Water Bay China

Numerous studies have shown that label noise can lead to poor generalization performance, negatively affecting classification accuracy. Therefore, understanding the effectiveness of classifiers trained using deep neural networks in the presence of noisy labels is of considerable practical significance. In this paper, we focus on the error bounds of excess risks for classification problems with noisy labels within deep learning frameworks. We begin by exploring loss functions with noise-tolerant properties, ensuring that the empirical minimizer on noisy data aligns with that on the true data. Next, we estimate the error bounds of the excess risks, expressed as a sum of statistical error and approximation error. We estimate the statistical error on a dependent (mixing) sequence, bounding it with the help of the associated independent block sequence. For the approximation error, we first express the classifiers as the composition of the softmax function and a continuous function from [0, 1]d to RK. The main task is then to estimate the approximation error for the continuous function from [0, 1]d to RK. Finally, we focus on the curse of dimensionality based on the low-dimensional manifold assumption. Copyright © 2025, The Authors. All rights reserved.

关键词： Risk assessment

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An automatic approach to develop the fourth-order and L2-stable lattice Boltzmann model for diagonal-anisotropic diffusion equations

arXiv

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arXiv 2025年

作者： Chen, Ying Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China The State Key Laboratory of Intelligent Manufacturing Equipment and Technology Huazhong University of Science and Technology Wuhan430074 China

This paper discusses how to develop a high-order multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the general d(≥ 1)-dimensional diagonal-anisotropic diffusion equation. Such an MRT-LB model considers the transformation matrix constructed in a natural way and the DdQ(2d2 + 1) [(2d2 + 1) discrete velocities in d-dimensional space] lattice structure. A key step in developing the high-order MRT-LB model is to determine the additional adjustable relaxation parameters and weight coefficients, which are used to eliminate the truncation errors at some certain orders of the MRT-LB model, while ensuring the stability of the MRT-LB model. In this work, we first present a unified MRT-LB model for the d-dimensional diagonal-anisotropic diffusion equation. Then, through the direct Taylor expansion, we analyze the macroscopic modified equations of the MRT-LB model up to fourth-order at the diffusive scaling, and further derive the conditions that ensure the MRT-LB model to be fourth-order consistent with the diagonal-anisotropic diffusion equation. In particular, when the diagonal-anisotropic diffusion equation is reduced to the isotropic type, we propose another MRT-LB model with the DdQ(2d+ 1) lattice structure [fewer discrete velocities than the DdQ(2d2 + 1) lattice structure], and the fourth-order conditions are similarly derived. Additionally, we also construct the fourth-order initialization scheme for the present LB method. After that, the condition which guarantees that the MRT-LB model can satisfy the stability structure is explicitly given, and we would like to point out that from a numerical perspective, once the stability structure is satisfied, the MRT-LB model must be L2 stable. In combination with the fourth-order consistent and L2 stability conditions, the relaxation parameters and weight coefficients of the MRT-LB model can be automatically given by a simple computer code. Finally, we perform numerical simulations of several benchmark problems, and f

关键词： Anisotropy

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THE GLOBAL ATTRACTOR FOR A VISCOUS WEAKLY DISSIPATIVE GENERALIZED TWO-COMPONENT μ-HUNTER-SAXTON SYSTEM

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Acta Mathematica Scientia 2018年第2期38卷 651-672页

作者：张磊刘斌 School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology

This article is concerned with the existence of global attractor of a weakly dissipative generalized two-component μ-Hunter-Saxton （gμHS2） system with viscous terms. Under the period boundary conditions and with the help of the Galerkin procedure and compactness method, we first investigate the existence of global solution for the viscous weakly dissipative （gμHS2） system. On the basis of some uniformly prior estimates of the solution to the viscous weakly dissipative （gμHS2） system, we show that the semi-group of the solution operator {S（t）}t≥0 has a bounded absorbing set. Moreover, we prove that the dynamical system {S（t）}t≥0 possesses a global attractor in the Sobolev space H2（S） × H2（S）.

关键词： Generalized two-component μ-Hunter-Saxton system viscous weakly dissipative existence global attractor period boundary conditions

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Backward Euler-Maruyama method applied to nonlinear hybrid stochastic differential equations with time-variable delay

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Science China Mathematics 2019年第3期62卷 597-616页

作者： Chengjian Zhang Ying Xie School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology

In this paper, we consider strong convergence and almost sure exponential stability of the backward Euler-Maruyama method for nonlinear hybrid stochastic differential equations with time-variable delay. Under the local Lipschitz condition and polynomial growth condition, it is proved that the backward Euler-Maruyama method is strongly convergent. Additionally, the moment estimates and almost sure exponential stability for the analytical solution are proved. Also, under the appropriate condition, we show that the numerical solutions for the backward Euler-Maruyama methods are almost surely exponentially stable. A numerical experiment is given to illustrate the computational effectiveness and the theoretical results of the method.

关键词： nonlinear hybrid stochastic differential equations time-variable delay backward Euler-Maruyama method strong convergence almost surely exponential stability

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Global Solvability for a Predator-Prey Model with Prey-Taxis and Rotational Flux Terms

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Chinese Annals of Mathematics,Series B 2024年第2期45卷 297-318页

作者： Guoqiang REN Bin LIU School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong Universityof Science and Technology

In this paper the author investigates the following predator-prey model with prey-taxis and rotational?ux terms■in a bounded domain with smooth *** presents the global existence of generalized solutions to the model... 详细信息

关键词： Predator-prey Prey-taxis Giopal existence Kotational flnux

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Hamilton-Jacobi method in reheating predictions for non-minimal coupling inflation

arXiv

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arXiv 2025年

作者： Zhang, Feng-Yi Chen, Li-Yang Zhai, Rongrong School of Mathematics and Physics University of South China Hengyang421001 China Hunan Key Laboratory of Mathematical Modeling and Scientific Computing University of South China Hengyang421001 China College of Physics and Engineering Technology Chengdu Normal University Sichuan Chengdu611130 China Department of Physics Xinzhou Normal University Shanxi Xinzhou034000 China

The Hamilton-Jacobi method offers a natural and concise framework for describing inflation, with implications that extend to the reheating phase. Additionally, reheating plays a crucial role in constraining the observationally viable parameter space of inflationary models. In this study, we employ the Hamilton-Jacobi approach to investigate reheating predictions within non-minimally coupled inflation models, comparing the metric and Palatini formulations. Our results show that the coupling effect suppresses the tensor-to-scalar ratio, aligning predictions with the Planck CMB and BICEP/Keck data in both formulations. Additionally, reheating predictions in the Palatini formulation are more sensitive to coupling strength variations, leading to a stronger suppression of the tensor-to-scalar ratio. This highlights a key difference in reheating dynamics between the metric and Palatini formulations. Copyright © 2025, The Authors. All rights reserved.

关键词： Tensors

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Splitting Method for Stochastic Navier-Stokes Equations

arXiv

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arXiv 2025年

作者： Zhu, Jie Zhu, Yujun Ming, Ju Gunzburger, Max D. School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China Department of Scientific Computing Florida State University TallahasseeFL32304 United States

This paper investigates the two-dimensional stochastic steady-state Navier-Stokes(NS) equations with additive random noise. We introduce an innovative splitting method that decomposes the stochastic NS equations into a deterministic NS component and a stochastic equation. We rigorously analyze the proposed splitting method from the perspectives of equivalence, stability, existence and uniqueness of the solution. We also propose a modified splitting scheme, which simplified the stochastic equation by omitting its nonlinear terms. A detailed analysis of the solution properties for this modified approach is provided. Additionally, we discuss the statistical errors with both the original splitting format and the modified scheme. Our theoretical and numerical studies demonstrate that the equivalent splitting scheme exhibits significantly enhanced stability compared to the original stochastic NS equations, enabling more effective handling of nonlinear characteristics. Several numerical experiments were performed to compare the statistical errors of the splitting method and the modified splitting method. Notably, the deterministic NS equation in the splitting method does not require repeated solving, and the stochastic equation in the modified scheme is free of nonlinear terms. These features make the modified splitting method particularly advantageous for large-scale computations, as it significantly improves computational efficiency without compromising accuracy. Copyright © 2025, The Authors. All rights reserved.

关键词： Navier Stokes equations

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Boundary Control of Coupled Wave Systems with Spatially-Varying Coefficients

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Journal of Systems Science & Complexity 2022年第4期35卷 1310-1329页

作者： FENG Xiaodan ZHANG Zhifei School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan430074China

This paper considers the stabilization of the coupled wave systems with spatially-varying *** authors design a state feedback controller by backstepping *** contrast to the previous work in the literature,the kernel equations become more complicated and the main difficulty lies in proving the existence and uniqueness of the solution to the kernel ***,using the backstepping approach,the authors verify the kernel equations,which is a system of coupled hyperbolic equations with spatially-varying ***,the existence and uniqueness of the kernel matrices is ***,the authors use a Lyapunov function to get the exponential stabilization of the closed-loop system.A numerical example is presented to illustrate the effectiveness of the proposed controller.

关键词： Backstepping boundary control coupled wave systems exponential stabilization spatially-varying coefficients

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Stabilization and Blow-up in an Infinite Memory Wave Equation with Logarithmic Nonlinearity and Acoustic Boundary Conditions

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Journal of Systems Science & Complexity 2024年第4期37卷 1368-1391页

作者： PENG Qingqing ZHANG Zhifei School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China

In this paper,the authors consider the stabilization and blow up of the wave equation with infinite memory,logarithmic nonlinearity and acoustic boundary *** authors discuss the existence of global solutions for the initial energy less than the depth of the potential well and investigate the energy decay estimates by introducing a Lyapunov ***,the authors establish the finite time blow up results of solutions and give the blow up time with upper bounded initial energy.

关键词： Acoustic boundary condition finite time blow-up general decay infinite memory logarithmic nonlinearity

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