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MODERATE DEVIATION PRINCIPLES FOR A REACTION DIFFUSION MODEL IN NON-EQUILIBRIUM

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arXiv 2024年

作者： Zhao, Linjie School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

We study moderate deviations from hydrodynamic limits of a reaction diffusion model. The process is defined as the superposition of the symmetric exclusion process with a Glauber dynamics. When the process starts from a product measure with a constant density, which is a non-equilibrium measure for the process, we prove that the re-scaled density fluctuation field satisfies the moderate deviation principle. Our proof relies on the so-called main lemma developed by Jara and Menezes [5, 6]. © 2024, CC BY.

关键词： Equilibrium constants

REQUIRED NUMBER OF ITERATIONS FOR SPARSE SIGNAL RECOVERY VIA ORTHOGONAL LEAST SQUARES

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Journal of Computational Mathematics 2023年第1期41卷 1-17页

作者： Haifeng Li Jing Zhang Jinming Wen Dongfang Li Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control College of Mathematics and Information ScienceHenan Normal UniversityXinxiang 453007China College of Information Science and Technology Jinan UniversityGuangzhou 510632China 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 countless applications,we need to reconstruct a K-sparse signal x∈R n from noisy measurements y=Φx+v,whereΦ∈R^(m×n)is a sensing matrix and v∈R m is a noise *** least squares(OLS),which selects at each step the column that results in the most significant decrease in the residual power,is one of the most popular sparse recovery *** this paper,we investigate the number of iterations required for recovering x with the OLS *** show that OLS provides a stable reconstruction of all K-sparse signals x in[2.8K]iterations provided thatΦsatisfies the restricted isometry property(RIP).Our result provides a better recovery bound and fewer number of required iterations than those proposed by Foucart in 2013.

关键词： Sparse signal recovery Orthogonal least squares(OLS) Restricted isometry property(RIP)

BOUNDARY VALUE METHODS FOR CA PUTO FR ACTIONAL DIFFERENTIAL EQUATIONS

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Journal of Computational Mathematics 2021年第1期39卷 108-129页

作者： Yongtao Zhou Chengiian Zhang Huiru Wang 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

This paper deals with the numerical computation and analysis for Caputo fractional differential equations(CFDEs).By combining the p-order boundary value methods(B-VMs)and the m-th Lagrange interpolation,a type of exte... 详细信息

关键词： Fractional differential equations Caputo derivatives Boundary value methods Local stability Unique solvability Convergence.

RANDOM PERTURBATIONS FOR THE CHEMOTAXIS-FLUID MODEL WITH FRACTIONAL DISSIPATION: GLOBAL PATHWISE WEAK SOLUTIONS

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arXiv 2023年

作者： Zhang, Lei Liu, Bin School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China

This paper considers a stochastically perturbed Keller-Segel-Navier-Stokes (KS-SNS) system arising from the biomathematics in two dimensions, where the diffusion of fluid is expressed by a fractional Laplacian with an exponent in [1/2, 1]. Our main result demonstrates that, under appropriate assumptions, the Cauchy problem of the KS-SNS system has a unique global probabilistically strong and analytically weak solution, which also confirms that the quadratic logistic source authentically contributes to the global existence of solutions. First, a three-layer approximate system is introduced, this system seems to be new in the studying of chemotaxis-fluid model and it enables one to construct approximate solutions in regular Hilbert spaces Hs(R2). Second, to accomplish the convergence progressively, a series of crucial entropy-energy inequalities for the approximations are derived, whose derivation strongly depends on the fine structure of the system and requires a novel strategy to adapt to the appearance of fractional dissipation and the unboundedness of domain. Third, based on these uniform bounds, we establish the existence of global martingale weak solutions by virtue of a stochastic compactness method. Finally, by applying the Littlewood-Paley theory, we show that the pathwise uniqueness holds for these martingale solution. As a by-product, a few interpolation inequalities in Besov spaces are obtained, which seems to be new and may have their own interest. Copyright © 2023, The Authors. All rights reserved.

关键词： Stochastic systems

The Exploration of Neural Collapse under Imbalanced Data

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arXiv 2024年

作者： Liu, Haixia 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

Neural collapse, a newly identified characteristic, describes a property of solutions during model training. In this paper, we explore neural collapse in the context of imbalanced data. We consider the L-extended unconstrained feature model with a bias term and provide a theoretical analysis of global minimizer. Our findings include: (1) Features within the same class converge to their class mean, similar to both the balanced case and the imbalanced case without bias. (2) The geometric structure is mainly on the left orthonormal transformation of the product of L linear classifiers and the right transformation of the class-mean matrix. (3) Some rows of the left orthonormal transformation of the product of L linear classifiers collapse to zeros and others are orthogonal, which relies on the singular values of Ŷ = (IK − 1/Nn1TK)D, where K is class size, n is the vector of sample size for each class, D is the diagonal matrix whose diagonal entries are given by √n. Similar results are for the columns of the right orthonormal transformation of the product of class-mean matrix and D. (4) The i-th row of the left orthonormal transformation of the product of L linear classifiers aligns with the i-th column of the right orthonormal transformation of the product of class-mean matrix and D. (5) We provide the estimation of singular values about Ŷ . Our numerical experiments support these theoretical *** Codes ***, *** Copyright © 2024, The Authors. All rights reserved.

关键词： Linear transformations

On the deterioration of convergence rate of spectral differentiations for functions with singularities

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arXiv 2025年

作者： Wang, Haiyong School of Mathematics and Statistics 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

Spectral differentiations are basic ingredients of spectral methods. In this work, we analyze the pointwise rate of convergence of spectral differentiations for functions containing singularities and show that the deteriorations of the convergence rate at the endpoints, singularities and other points in the smooth region exhibit different patterns. As the order of differentiation increases by one, we show for functions with an algebraic singularity that the convergence rate of spectral differentiation by Jacobi projection deteriorates two orders at both endpoints and only one order at each point in the smooth region. The situation at the singularity is more complicated and the convergence rate either deteriorates two orders or does not deteriorate, depending on the parity of the order of differentiation, when the singularity locates in the interior of the interval and deteriorates two orders when the singularity locates at the endpoint. Extensions to some related problems, such as the spectral differentiation using Chebyshev interpolation, are also discussed. Our findings justify the error localization property of Jacobi approximation and differentiation and provide some new insight into the convergence behavior of Jacobi spectral *** Codes 65D25, 41A10, 41A25 Copyright © 2025, The Authors. All rights reserved.

关键词： Chebyshev approximation

Incomplete crossing and semi-topological horseshoes

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arXiv 2025年

作者： Cheng, Junfeng Yang, Xiao-Song School of Mathematics and Statistics 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

This paper enriches the topological horseshoe theory using finite subshift theory in symbolic dynamical systems, and develops an elementary framework addressing incomplete crossing and semi-horseshoes. Two illustrative examples are provided: one from the perturbed Duffing system and another from a polynomial system proposed by Chen, demonstrating the prevalence of semi-horseshoes in chaotic systems. Moreover, the semi-topological horseshoe theory enhances the detection of chaos and improves the accuracy of topological entropy estimation. Copyright © 2025, The Authors. All rights reserved.

关键词： Topology

Complex generalized Gauss-Radau quadrature rules for Hankel transforms of integer order

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arXiv 2024年

作者： Wang, Haiyong Wu, Menghan School of Mathematics and Statistics 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

Complex Gaussian quadrature rules for oscillatory integral transforms have the advantage that they can achieve optimal asymptotic order. However, their existence for Hankel transform can only be guaranteed when the order of the transform belongs to [0, 1/2]. In this paper we consider the construction of generalized Gauss-Radau quadrature rules for Hankel transform. We show that, if adding certain value and derivative information at the left endpoint, then complex generalized Gauss-Radau quadrature rules for Hankel transform of integer order can be constructed with theoretical guarantees. Orthogonal polynomials that are closely related to such quadrature rules are investigated and their existence for even degrees is proved. Numerical experiments are presented to confirm our *** Codes 65R10, 65D32 © 2024, CC BY-NC-ND.

关键词： Integral equations

Stochastic homogenization of nonlinear evolution equations with space-time nonlocality

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arXiv 2023年

作者： Chen, Junlong Tang, Yanbin 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

In this paper we consider the homogenization problem of nonlinear evolution equations with space-time non-locality, the problems are given by Beltritti and Rossi [JMAA, 2017, 455: 1470-1504]. When the integral kernel J(x,t;y,s) is re-scaled in a suitable way and the oscillation coefficient ν(x,t;y,s) possesses periodic and stationary structure, we show that the solutions uΕ(x,t) to the perturbed equations converge to u0(x,t), the solution of corresponding local nonlinear parabolic equation as scale parameter Ε → 0+. Then for the nonlocal linear index p = 2 we give the convergence rate such that (formula presented). Furthermore, we obtain that the normalized difference (formula presented) converges to a solution of an SPDE with additive noise and constant coefficients. Finally, we give some numerical formats for solving non-local space-time homogenization. Copyright © 2023, The Authors. All rights reserved.

关键词： Additive noise