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Invariant foliations for stochastic dynamical systems with multiplicative stable Lévy noise

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arXiv 2018年

作者： Chao, Ying Wei, Pingyuan Yuan, Shenglan School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China

This work deals with the dynamics of a class of stochastic dynamical systems with a multiplicative non-Gaussian Lévy noise. We first establish the existence of stable and unstable foliations for this kind of system via the Lyapunov-Perron method. Then we examine the geometric structure of the invariant foliations, and their relation with invariant manifolds. Finally, we illustrate our results in an example. Copyright © 2018, The Authors. All rights reserved.

关键词： Differential equations

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

A Diffuse-Interface Lattice Boltzmann Method for the Dendritic Growth with Thermosolutal Convection

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Communications in Computational Physics 2023年第4期33卷 1164-1188页

作者： Chengjie Zhan Zhenhua Chai Baochang Shi Ping Jiang Shaoning Geng Dongke Sun School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan430074P.R.China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and TechnologyWuhan 430074P.R.China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan430074P.R.China The State Key Laboratory of Digital Manufacturing Equipment and Technology School of Mechanical Science and EngineeringHuazhong University of Science and TechnologyWuhan 430074P.R.China School of Mechanical Engineering Southeast UniversityNanjing 211189P.R.China

In this work,we proposed a diffuse-interface model for the dendritic growth with thermosolutal *** this model,the sharp boundary between the fluid and solid dendrite is firstly replaced by a thin but nonzero thickness diffuse interface,which is described by the order parameter,and the diffuse-interface based governing equations for the dendritic growth are *** solve the model for the dendritic growth with thermosolutal convection,we also developed a diffuse-interface multirelaxation-time lattice Boltzmann(LB)*** this method,the order parameter in the phase-field equation is combined into the force caused by the fluid-solid interaction,and the treatment on the complex fluid-solid interface can be *** addition,four LB models are designed for the phase-field equation,concentration equation,temperature equation and the Navier-Stokes equations in a unified ***,we performed some simulations of the dendritic growth to test the present diffuse-interface LB method,and found that the numerical results are in good agreements with some previous works.

关键词： Dendritic growth diffuse interface lattice Boltzmann method phase-field method.

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

Open-loop and closed-loop solvabilities for discrete-time mean-field stochastic linear quadratic optimal control problems

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arXiv 2023年

作者： Song, Teng Liu, Bin Department of Mathematics Wuhan University of Technology China 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 discusses the discrete-time mean-field stochastic linear quadratic optimal control problems, whose weighting matrices in the cost functional are not assumed to be definite. The open-loop solvability is characterized by the existence of the solution to a mean-field forward-backward stochastic difference equations with a convexity condition and a stationary condition. The closed-loop solvability is presented by virtue of the existences of the regular solution to the generalized Riccati equations and the solution to the linear recursive equation, which is also shown by the uniform convexity of the cost functional. Moreover, based on a family of uniformly convex cost functionals, the finiteness of the problem is characterized. Also, it turns out that a minimizing sequence, whose convergence is equivalent to the open-loop solvability of the problem. Finally, some examples are given to illustrate the theory *** Codes 93E20, 93C55, 49N10 Copyright © 2023, The Authors. All rights reserved.

关键词： Riccati equations

Simulation of nitrogen transport in a surficial aquifer and estimation of nitrogen load from septic systems in the Indian River Lagoon Area, Florida

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Simulation of nitrogen transport in a surficial aquifer and ...

World Environmental and Water Resources Congress 2015: Floods, Droughts, and Ecosystems

作者： Sayemuzzaman, Mohammad Ye, Ming Zhu, Yun Hicks, Richard Shi, Liangsheng Department of Scientific Computing Florida State University United States State Key Laboratory of Water Resources and Hydropower Engineering Science Wuhan University Wuhan Hubei China Florida Department of Environmental Protection TallahasseeFL United States

ISBN: (数字)9780784479162

ISBN: (纸本)9780784479162

Onsite sewage treatment and disposal system (OSTDS) or septic systems, is one of the significant sources of nitrogen in groundwater and surface water, poseing serious threats to human health and the environment. The ArcGIS-based Nitrate Load Estimation Toolkit (ArcNLET) has been used to simulate nitrogen transport and estimate nitrogen load from septic systems to surface waterbodies through the surficial aquifer. It is based on a simplified conceptual model of groundwater flow and nitrogen transport. This study utilizes the ArcNLET tool for the Main-South Canal (MSC) drainage basins of Indian River County (IRC) area which discharge to the Indian River Lagoon (IRL), FL. The purpose of this study is to estimate the nitrogen load from 12,735 septic systems to the IRL and other surface waterbodies in MSC drainage basins. This study found, all the waterbodies(574 in number) including the IRL have been receiving a total nitrogen load of 13,742 lbs/yr. Twenty out of the 574 watebodies receive 50.5% (6,802 lbs/yr) of the total load. Among the 12,735 septic systems, only 358 septic systems contributed nitrogen load directly to the IRL, which is 2.01% (271 lbs/yr) of the total load. It is also found that the estimated ground water load of nitrogen to the southern drainage basin is greater than the estimated septic tank load to the main drainage basin. The estimation of nitrogen load exhibits spatial variability, which is useful to facilitate decisions on the sewering of residential areas served by OSTDS to reduce nitrogen load to the canals and the IRL. The modeling results may change when new data are available for model calibration. © 2015 ASCE.

关键词： Aquifers

A unified framework for asymptotic analysis and computation of finite Hankel transform

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arXiv 2018年

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

In this paper we present a unified framework for asymptotic analysis and computation of the finite Hankel transform. This framework enables us to derive asymptotic expansions of the transform, including the cases where the oscillator has zeros and stationary points. As a consequence, two efficient and affordable methods for computing the transform numerically are developed and a detailed analysis of their asymptotic error estimate is carried out. Numerical examples are provided to confirm our analysis.65D30, 65D32, 65R10, 41A60 Copyright © 2018, The Authors. All rights reserved.

关键词： Asymptotic analysis

Gradient Descent for Symmetric Tensor Decomposition

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Annals of Applied Mathematics 2022年第4期38卷 385-413页

作者： Jian-Feng Cai Haixia Liu Yang Wang Department of Mathematics The Hong Kong University of Science and TechnologyClear Water BayKowloonHong KongChina School of Mathematics and Statistics&Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhanHubei 430074China

Symmetric tensor decomposition is of great importance in *** studies have employed a greedy approach,where the main idea is to first find a best rank-one approximation of a given tensor,and then repeat the process to the residual tensor by subtracting the rank-one *** this paper,we focus on finding a best rank-one approximation of a given orthogonally order-3 symmetric *** give a geometric landscape analysis of a nonconvex optimization for the best rank-one approximation of orthogonally symmetric *** show that any local minimizer must be a factor in this orthogonally symmetric tensor decomposition,and any other critical points are linear combinations of the ***,we propose a gradient descent algorithm with a carefully designed initialization to solve this nonconvex optimization problem,and we prove that the algorithm converges to the global minimum with high probability for orthogonal decomposable *** result,combined with the landscape analysis,reveals that the greedy algorithm will get the tensor CP low-rank *** results are provided to verify our theoretical results.

关键词： Gradient descent random initialization symmetric tensor decomposition CP decomposition linear convergence

Efficient Numerical Computation of Time-Fractional Nonlinear Schrodinger Equations in Unbounded Domain

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Communications in Computational Physics 2019年第1期25卷 218-243页

作者： Jiwei Zhang Dongfang Li Xavier Antoine Beijing Computational Science Research Center Beijing 100193P.R.China 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 Institut Elie Cartan de Lorraine UMR CNRS 7502Universit́e de LorraineInria Nancy-Grand EstSPHINX TeamF-54506 Vandoeuvre-l`es-Nancy CedexFrance

The aim of this paper is to derive a stable and efficient scheme for solving the one-dimensional time-fractional nonlinear Schrodinger equation set in an unbounded *** first derive absorbing boundary conditions for the fractional system by using the unified approach introduced in[47,48]and a linearization ***,the initial boundary-value problem for the fractional system with ABCs is discretized,a stability analysis is developed and the error estimate O(h^(2)+τ)is *** accel-erate the L1-scheme in time,a sum-of-exponentials approximation is introduced to speed-up the evaluation of the Caputo fractional *** resulting algorithm is highly efficient for long time ***,we end the paper by reporting some numerical simulations to validate the properties(accuracy and efficiency)of the derived scheme.

关键词： Time-fractional nonlinear Schrodinger equation absorbing boundary condition sta-bility analysis convergence analysis sum-of-exponentials approximation