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Revisiting the central limit theorems for the SGD-type methods 重温 SGD 类方法的中心极限定理
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2024-07-15 DOI: 10.4310/cms.2024.v22.n5.a10
Tiejun Li, Tiannan Xiao, Guoguo Yang
{"title":"Revisiting the central limit theorems for the SGD-type methods","authors":"Tiejun Li, Tiannan Xiao, Guoguo Yang","doi":"10.4310/cms.2024.v22.n5.a10","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n5.a10","url":null,"abstract":"We revisited the central limit theorem (CLT) for stochastic gradient descent (SGD) type methods, including the vanilla SGD, momentum SGD and Nesterov accelerated SGD methods with constant or vanishing damping parameters. By taking advantage of Lyapunov function technique and $L^p$ bound estimates, we established the CLT under more general conditions on learning rates for broader classes of SGD methods as compared to previous results. The CLT for the time average was also investigated, and we found that it held in the linear case, while it was not generally true in nonlinear situation. Numerical tests were also carried out to verify our theoretical analysis.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Representation theorem for multivariable totally symmetric functions 多变量完全对称函数的表示定理
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2024-07-15 DOI: 10.4310/cms.2024.v22.n5.a2
Chongyao Chen, Ziang Chen, Jianfeng Lu
{"title":"Representation theorem for multivariable totally symmetric functions","authors":"Chongyao Chen, Ziang Chen, Jianfeng Lu","doi":"10.4310/cms.2024.v22.n5.a2","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n5.a2","url":null,"abstract":"In this work, we establish a representation theorem for multivariable totally symmetric functions: a multisymmetric continuous function must be the composition of a continuous function and a set of generators of the multisymmetric polynomials. We then study the singularity and geometry of the generators, and show that the regularity may become worse after applying the decomposition.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Differentiable Hartman-Grobman Theorem via modulus of continuity: A sharp result on linearization in general Banach space 通过连续性模数的可微哈特曼-格罗布曼定理一般巴拿赫空间线性化的尖锐结果
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2024-07-15 DOI: 10.4310/cms.2024.v22.n5.a8
Zhicheng Tong, Yong Li
{"title":"Differentiable Hartman-Grobman Theorem via modulus of continuity: A sharp result on linearization in general Banach space","authors":"Zhicheng Tong, Yong Li","doi":"10.4310/cms.2024.v22.n5.a8","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n5.a8","url":null,"abstract":"As is well known the classical Hartman–Grobman theorem states that a $C^1$ mapping can be $C^0$ linearized near its hyperbolic fixed point in $mathbb{R}^n$. However, it is quite nontrivial to guarantee the local homeomorphism to be differentiable. Recently, the regularity assumption on derivative of the mapping has been weakened to Hölder’s type, significantly improving the work of $C^infty$, but still unknown for only differentiable case. We will try to touch this question in this paper. Without Hölder’s type, we first consider the existence and regularity of weak-stable manifolds for homeomorphisms with contraction in a Banach space, and further study linearization of mappings near hyperbolic fixed points. More precisely, we propose an Integrability Condition for regularity on linearization which is proved to be sharp, and establish a differentiable Hartman–Grobman theorem via modulus of continuity in a general Banach space. Thus we provide an almost complete answer to the question mentioned above.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Chemotactic reaction enhancement in one dimension 一维的趋化反应增强
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2024-07-15 DOI: 10.4310/cms.2024.v22.n5.a5
Yishu Gong, Alexander Kiselev
{"title":"Chemotactic reaction enhancement in one dimension","authors":"Yishu Gong, Alexander Kiselev","doi":"10.4310/cms.2024.v22.n5.a5","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n5.a5","url":null,"abstract":"Chemotaxis, which involves the directed movement of cells in response to a chemical gradient, plays a crucial role in a broad variety of biological processes. Examples include bacterial motion, the development of single-cell or multicellular organisms, and immune responses. Chemotaxis directs bacteria’s movement to find food (e.g., glucose) by swimming toward the highest concentration of food molecules. In multicellular organisms, chemotaxis is critical to early development (e.g., movement of sperm towards the egg during fertilization). Chemotaxis also helps mobilize phagocytic and immune cells at sites of infection, tissue injury, and thus facilitates immune reactions. In this paper, we study a PDE system that describes chemotactic processes in one dimension, which may correspond to a thin channel, the setting relevant in many applications: for example, spermatozoa progression to the ovum inside a Fallopian tube or immune response in a blood vessel. Our objective is to obtain qualitatively precise estimates on how chemotaxis improves reaction efficiency, when compared to purely diffusive situation. The techniques we use to achieve this goal include a variety of comparison principles and analysis of mass transport for a class of Fokker–Planck operators.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Non-uniqueness of transonic shock solutions to Euler–Poisson system with varying background charges 背景电荷变化的欧拉-泊松系统跨音速冲击解的非唯一性
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2024-03-04 DOI: 10.4310/cms.2024.v22.n3.a7
Ben Duan, Haoran Zheng
{"title":"Non-uniqueness of transonic shock solutions to Euler–Poisson system with varying background charges","authors":"Ben Duan, Haoran Zheng","doi":"10.4310/cms.2024.v22.n3.a7","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n3.a7","url":null,"abstract":"The Euler–Poisson equations with varying background charges in finitely long flat nozzles are investigated, for which two and only two transonic shock solutions are constructed. In href{https://dx.doi.org/10.4310/CMS.2012.v10.n2.a1}{[textrm{T. Luo and Z.P. Xin, Commun. Math. Sci., 10:419–462, 2012}]}, Luo and Xin established the wellposedness of steady Euler–Poisson equations for the constant background charge. Motivated by their pioneering work and combined with the special physical character of semiconductor devices, we propose the transonic shock problem in which the density of the background charge is a piecewise constant function and its discontinuity is determined only by shock fronts. The existence and non-uniqueness of transonic shock solutions are obtained via the method of shock matching.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034367","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Hydrodynamic traffic flow models including random accidents: A kinetic derivation 包括随机事故在内的水动力交通流模型：动力学推导
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2024-03-04 DOI: 10.4310/cms.2024.v22.n3.a10
Felisia Angela Chiarello, Simone Göttlich, Thomas Schillinger, Andrea Tosin
{"title":"Hydrodynamic traffic flow models including random accidents: A kinetic derivation","authors":"Felisia Angela Chiarello, Simone Göttlich, Thomas Schillinger, Andrea Tosin","doi":"10.4310/cms.2024.v22.n3.a10","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n3.a10","url":null,"abstract":"We present a formal kinetic derivation of a second order macroscopic traffic model from a stochastic particle model. The macroscopic model is given by a system of hyperbolic partial differential equations (PDEs) with a discontinuous flux function, in which the traffic density and the headway are the averaged quantities. A numerical study illustrates the performance of the second order model compared to the particle approach. We also analyse numerically uncertain traffic accidents by considering statistical measures of the solution to the PDEs.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Autonomous vehicles driving traffic: The Cauchy problem 自动驾驶汽车在交通中行驶：考奇问题
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2024-03-04 DOI: 10.4310/cms.2024.v22.n3.a9
Mauro Garavello, Francesca Marcellini
{"title":"Autonomous vehicles driving traffic: The Cauchy problem","authors":"Mauro Garavello, Francesca Marcellini","doi":"10.4310/cms.2024.v22.n3.a9","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n3.a9","url":null,"abstract":"This paper deals with the Cauchy problem for a PDE‑ODE model, where a system of two conservation laws, namely the Two-Phase macroscopic model proposed in [Rinaldo M. Colombo, Francesca Marcellini, and Michel Rascle, $href{https://doi.org/10.1137/090752468}{textrm{SIAM J. Appl. Math., 70(7):2652–2666, 2010}}$], is coupled with an ordinary differential equation describing the trajectory of an autonomous vehicle (AV), which aims to control the traffic flow. Under suitable assumptions, we prove a global-in-time existence result.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Analysis and computation for the scattering problem of electromagnetic waves in chiral media 手性介质中电磁波散射问题的分析与计算
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2024-03-04 DOI: 10.4310/cms.2024.v22.n3.a5
Gang Bao, Lei Zhang
{"title":"Analysis and computation for the scattering problem of electromagnetic waves in chiral media","authors":"Gang Bao, Lei Zhang","doi":"10.4310/cms.2024.v22.n3.a5","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n3.a5","url":null,"abstract":"This paper considers an obstacle scattering problem in a chiral medium under circularly polarized oblique plane wave incidence, which can be represented as a combination of a left-circularly polarized plane wave and a right-circularly polarized one. We apply a reduced model problem with coupled oblique derivative boundary conditions, describing the cross-coupling effect of electric and magnetic fields. A novel boundary integral equation is constructed by introducing single-layer potential operators and the corresponding normal and tangential derivative operators. The corresponding properties are obtained by splitting techniques to overcome the singularity of integral operators. A numerical method for solving the boundary integral equation is developed, whose convergence is proved. Numerical results are presented to show the performance of the proposed method.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Improved uniform error bounds of an exponential wave integrator method for the Klein–Gordon–Schrödinger equation with the small coupling constant 具有小耦合常数的克莱因-戈登-薛定谔方程指数波积分器方法的改进均匀误差边界
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2024-03-04 DOI: 10.4310/cms.2024.v22.n3.a1
Jiyong Li
{"title":"Improved uniform error bounds of an exponential wave integrator method for the Klein–Gordon–Schrödinger equation with the small coupling constant","authors":"Jiyong Li","doi":"10.4310/cms.2024.v22.n3.a1","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n3.a1","url":null,"abstract":"Recently, the long-time numerical simulation and error analysis of PDEs with weak nonlinearity (or small potentials) become an interesting topic. However, the existing results of long-time error analysis mostly focus on the single equations. In this paper, for the Klein–Gordon–Schrödinger equation (KGSE) with a small coupling constant $varepsilon in (0,1]$, we propose an exponential wave integrator Fourier pseudo-spectral (EWIFP) method by reformulating the KGSE into a coupled nonlinear Schrödinger system (CNLSS). Through careful and rigorous analysis, we establish improved error bounds for the numerical solution at $O(h^m + varepsilon tau^2)$ in the long-time domain up to $O(1/varepsilon)$ where $m$ is determined by the regularity conditions, h is the mesh size and τ is the time step, respectively. Compared with the existing results, our analysis shows the long-time errors of numerical solution for the KGSE. In error analysis, in addition to the classical tools such as energy method and cut-off technique, we also adopt the regularity compensation oscillation (RCO) technique which has been developed recently to analyze the accumulation of errors carefully. The numerical experiments support our error estimates and demonstrate the long-term stability of discrete mass and energy. To the best of our knowledge, there has not been any relevant long-time error analysis for the KGSE and any improved uniform error bounds for an exponential wave integrator. Our work is novel and provides a reference for analyzing the improved error bounds of the numerical methods for other coupled equations.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140035267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Global existence and asymptotic behavior of the full Euler system with damping and radiative effects in $mathbb{R}^3$ 在 $mathbb{R}^3$ 中具有阻尼和辐射效应的全欧拉系统的全局存在性和渐近行为
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2024-03-04 DOI: 10.4310/cms.2024.v22.n3.a8
Shijin Deng, Wenjun Wang, Feng Xie, Xiongfeng Yang
{"title":"Global existence and asymptotic behavior of the full Euler system with damping and radiative effects in $mathbb{R}^3$","authors":"Shijin Deng, Wenjun Wang, Feng Xie, Xiongfeng Yang","doi":"10.4310/cms.2024.v22.n3.a8","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n3.a8","url":null,"abstract":"In this paper, we study the global existence and the large-time behavior of solutions to the Cauchy problem of the full Euler system with damping and radiative effects around some constant equilibrium states. It is well-known that the solutions may blow up in finite time without the additional damping and radiative effects, and the global existence of the solutions obtained in this paper shows that these two effects together prevent the formation of the singularity when the initial perturbation is small. Combining the Green’s function method and energy estimates, we consider the pointwise structures of the solutions to obtain a precise description of the system. The construction of the Green’s function includes three steps: singularity removal, long wave-short wave decomposition and weighted energy estimate. Finally, we achieve the pointwise estimates of the solutions in the small perturbation framework by Duhamel’s principle, the pointwise structure of the Green’s function established for the linearized equations and bounded estimates for higher order derivatives of the solutions together.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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