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Convergence analysis of Laguerre approximations for analytic functions

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arXiv 2023年

Laguerre spectral approximations play an important role in the development of efficient algorithms for problems in unbounded domains. In this paper, we present a comprehensive convergence rate analysis of Laguerre spectral approximations for analytic functions. By exploiting contour integral techniques from complex analysis, we prove that Laguerre projection and interpolation methods of degree n converge at the root-exponential rate O(exp(−2ρ√n)) with ρ > 0 when the underlying function is analytic inside and on a parabola with focus at the origin and vertex at z = −ρ2. As far as we know, this is the first rigorous proof of root-exponential convergence of Laguerre approximations for analytic functions. Several important applications of our analysis are also discussed, including Laguerre spectral differentiations, Gauss-Laguerre quadrature rules, the scaling factor and the Weeks method for the inversion of Laplace transform, and some sharp convergence rate estimates are derived. Numerical experiments are presented to verify the theoretical *** Codes 41A25, 41A10 © 2023, CC BY-NC-SA.

关键词： Laplace transforms

Compensated projected Euler method for stochastic differential equations with jumps under global monotonicity condition

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arXiv 2018年

作者： Li, Min Huang, Chengming 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 presents and analyzes the compensated projected Euler-Maruyama method for stochastic differential equations with jumps under a global monotonicity condition. Compared with existing conditions, this condition allows the jump-diffusion coefficient to be growth superlinearly. Moreover, the method is proved to be convergent with strongly order 1/2 on the discrete time level. Finally, some numerical experiments are carried out to confirm the theoretical results. Copyright © 2018, The Authors. All rights reserved.

关键词： Differential equations

Carleman and observability estimates for stochastic beam equation

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arXiv 2017年

作者： Zhen, Maoding He, Jinchun Xu, Haoyuan Yang, Meihua 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

In this paper, we establish a weight identity for stochastic beam equation by means of the multiplier method. Based on this identity, we first establish the global Carleman estimate for the special system with zero initial value and end value, then a revised Carleman estimate for stochastic beam equation is established through a cutoff technique. Finally, we use the revised Carleman estimate to get the required boundary observability estimate. Copyright © 2017, The Authors. All rights reserved.

关键词： Stochastic systems

Three-stage GA-LM algorithm for surface roughness parameters formula construction

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Jisuanji Fuzhu Sheji Yu Tuxingxue Xuebao/Journal of Computer-Aided Design and Computer Graphics 2015年第11期27卷 2192-2200页

作者： Liu, Xinru Liu, Shengjun Tang, Jinyuan Zhou, Wei Institute of Engineering Modeling and Scientific Computing School of Mathematics and Statistics Central South University Changsha410083 China State Key Laboratory of High Performance Complex Manufacturing Central South University Changsha410083 China

Based on the function approximation theory, the formulas for calculating surface roughness parameters are obtained by optimizing the coefficients of the function template. At first, a global approximate solution set is obtained by optimizing the initial function template using genetic algorithm. Then, the set is taken as the initial value of Levenberg-Marquardt (LM) algorithm to obtain the better local optimal solution set, and the two algorithms are used in turn until the solutions are convergent or the largest switching times are reached. At last, according to the convergence precision of the solutions and the value of each monomial, the growth or trim operation is conducted on the function template, and the two optimization algorithms are executed again;until the termination conditions are satisfied. The numerical simulation examples show the algorithm has the better ability to find the optimal solution and the good robustness. Moreover, the method is easy to carry out, and can be used in engineering practice. ©, 2015, Institute of computing Technology. All right reserved.

关键词： Genetic algorithms

Discrete unified gas kinetic scheme for nonlinear convection-diffusion equations

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Physical Review E 2020年第2期101卷 023306-023306页

作者： Jinlong Shang Zhenhua Chai Huili Wang Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China School of Mathematics and Computer Science Wuhan Textile University Wuhan 430073 China

In this paper, we develop a discrete unified gas kinetic scheme (DUGKS) for a general nonlinear convection-diffusion equation (NCDE) and show that the NCDE can be recovered correctly from the present model through the Chapman-Enskog analysis. We then test the present DUGKS through some classic convection-diffusion equations, and we find that the numerical results are in good agreement with analytical solutions and that the DUGKS model has a second-order convergence rate. Finally, as a finite-volume method, the DUGKS can also adopt the nonuniform mesh. Besides, we perform some comparisons among the DUGKS, the finite-volume lattice Boltzmann model (FV-LBM), the single-relaxation-time lattice Boltzmann model (SLBM), and the multiple-relaxation-time lattice Boltzmann model (MRT-LBM). The results show that the present DUGKS is more accurate than the FV-LBM, more stable than the SLBM, and almost has the same accuracy as the MRT-LBM. Moreover, the use of nonuniform mesh may make the DUGKS model more flexible.

关键词： Lattice-Boltzmann methods

Density functions for the overdamped generalized Langevin equation and its Euler-Maruyama method: smoothness and convergence

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arXiv 2024年

作者： Dai, Xinjie Jin, Diancong School of Mathematics and Statistics Yunnan University Kunming650504 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

This paper focuses on studying the convergence rate of the density function of the Euler- Maruyama (EM) method, when applied to the overdamped generalized Langevin equation with fractional noise which serves as an important model in many fields. Firstly, we give an improved upper bound estimate for the total variation distance between random variables by their Malliavin-Sobolev norms. Secondly, we establish the existence and smoothness of the density function for both the exact solution and the numerical one. Based on the above results, the convergence rate of the density function of the numerical solution is obtained, which relies on the regularity of the noise and kernel. This convergence result provides a powerful support for numerically capturing the statistical information of the exact solution through the EM method. Copyright © 2024, The Authors. All rights reserved.

关键词： Differential equations

Improved phase-field-based lattice Boltzmann method for thermocapillary flow

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Physical Review E 2022年第1期105卷 015314-015314页

作者： Liqing Yue Zhenhua Chai Huili Wang Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China School of Mathematical and Computer Sciences Wuhan Textile University Wuhan 430074 China

In this paper, we present an improved phase-field-based lattice Boltzmann (LB) method for thermocapillary flows with large density, viscosity, and thermal conductivity ratios. The present method uses three LB models to solve the conservative Allen-Cahn equation, the incompressible Navier-Stokes equations, and the temperature equation. To overcome the difficulty caused by the convection term in solving the convection-diffusion equation for the temperature field, we first rewrite the temperature equation as a diffuse equation where the convection term is regarded as the source term and then construct an improved LB model for the diffusion equation. The macroscopic governing equations can be recovered correctly from the present LB method; moreover, the present LB method is much simpler and more efficient. In order to test the accuracy of this LB method, several numerical examples are considered, including the planar thermal Poiseuille flow of two immiscible fluids, the two-phase thermocapillary flow in a nonuniformly heated channel, and the thermocapillary Marangoni flow of a deformable bubble. It is found that the numerical results obtained from the present LB method are consistent with the theoretical prediction and available numerical data, which indicates that the present LB method is an effective approach for the thermocapillary flows.

关键词： Drops & bubbles Multiphase flows Lattice-Boltzmann methods Navier-Stokes equation

GAUSSIAN QUADRATURE RULES FOR COMPOSITE HIGHLY OSCILLATORY INTEGRALS

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arXiv 2021年

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

Highly oscillatory integrals of composite type arise in electronic engineering and their calculations are a challenging problem. In this paper, we propose two Gaussian quadrature rules for computing such integrals. The first one is constructed based on the classical theory of orthogonal polynomials and its nodes and weights can be computed efficiently by using tools of numerical linear algebra. An interesting connection between the quadrature nodes and the Legendre points is proved and it is shown that the rate of convergence of this rule depends solely on the regularity of the non-oscillatory part of the integrand. The second one is constructed with respect to a sign-changing function and the classical theory of Gaussian quadrature can not be used anymore. We explore theoretical properties of this Gaussian quadrature, including the trajectories of the quadrature nodes and the convergence rate of these nodes to the endpoints of the integration interval, and prove its asymptotic error estimate under suitable hypotheses. Numerical experiments are presented to demonstrate the performance of the proposed *** Codes 65D30 Copyright © 2021, The Authors. All rights reserved.

关键词： Computation theory

SAMPLE PATH MDP FOR THE CURRENT AND THE TAGGED PARTICLE IN THE SSEP

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arXiv 2023年

作者： Xue, Xiaofeng Zhao, Linjie School of Mathematics and Statistics Beijing Jiaotong University Beijing100044 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

We prove sample path moderate deviation principles (MDP) for the current and the tagged particle in the symmetric simple exclusion process, which extends the results in [39], where the MDP was only proved at any fixed...

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