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Numerical Methods For Fractional Variational Problems Depending On Indefinite Integrals

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Numerical Methods For Fractional Variational Problems Depend...

第五届国际自动控制联合会分数阶导数及其应用会议

作者： Dongling Wang Aiguo Xiao Hunan Key Laboratory for Computation and Simulation in Scienceand Engineering Key Laboratory of Intelligent Computing &Information Processing of Ministry of Education University XiangtanHunan P.R.China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education University Xiangtan Hunan P.R.China

In this paper,the fractional variational integrators for fractional variational problems depending on indefinite integrals in terms of Caputo derivative are *** corresponding fractional discrete Euler-Lagrange equatio... 详细信息

关键词： Numerical Methods For Fractional Variational Problems Depending On Indefinite Integrals

Maximum-norm error analysis of a difference scheme for the space fractional CNLS

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Applied Mathematics and computation 2015年 257卷 241-251页

作者： Dongling Wang Aiguo Xiao Wei Yang Department of Mathematics and Center for Nonlinear Studies Northwest University Xi’an Shaanxi 710127 PR China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan 411105 PR China

The difference method for the space fractional coupled nonlinear Schrödinger equations (CNLS) is studied. The fractional centered difference is used to approximate the space fractional Laplacian. This scheme conserves the discrete mass and energy. Due to the nonlocal nature of fractional Laplacian, in the classic Sobolev space, it is hard to obtain the error estimation in l ∞ . To overcome this difficulty, the fractional Sobolev space H α / 2 and a fractional norm equivalence in H α / 2 are introduced. Then the convergence of order O ( h 2 + τ 2 ) in l ∞ is proved by fractional Sobolev inequality, where h is the mesh size and τ is the time step. Numerical examples are given to illustrate the theoretical results at last.

关键词： Fractional Schrödinger equations Fractional centered difference Convergence analysis

Elliptical characteristics and numerical simulation of the PSEs of ferrofluid

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Elliptical characteristics and numerical simulation of the P...

中国力学大会（CCTAM2019）

作者： Li Mingjun Tang Shujiang Hiroshi Yamaguchi Jie Wu School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Department of Mechanical Engineering Doshisha University

Ferrofluid is a colloidal suspension system with magnetic nanoparticles, the overall field of study in ferrofluid has a highly interdisciplinary character in practical applications. In this paper a set of parabolized stability equations(PSEs)of ferrofluid is represented by Rosensweig equations. The characteristic analysis show that in subsonic region the nonlinear PSEs of ferrofluid are parabolical, and in supersonic region are elliptical. The PSEs of ferrofluid is solved numerically for heat conduction process and RT instability problem, respectively. The results of characteristic analysis and numerical simulation all verify that the basic characteristic of ferrofluid are not influenced by the external magnetic field, and the viscosity is the only reason for the variation of ellipticity of PSEs of ferrofluid. That is to say, the external magnetic field and the viscosity are essentially different.

关键词： Ferrofluid Magnetic nanoparticles Parabolized stability equations Ellipticity

Two-level additive preconditioners for edge element discretizations of time-harmonic Maxwell equations

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Computers & Mathematics with Applications 2013年第4期66卷 432-440页

作者： Liuqiang Zhong Chunmei Liu Shi Shu School of Mathematical Sciences South China Normal University Guangzhou 510631 China Department of Mathematics and Computational Science Hunan University of Science and Engineering Yongzhou 425199 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan 411105 China

Two-level additive preconditioners are presented for edge element discretizations of time-harmonic Maxwell equations. The key is to construct a special “coarse mesh” space, which adds the kernel of the curl -operator in a fine space to a coarse mesh space, to solve the original problem, and then uses the fine mesh space to solve the H ( curl ) -elliptic problem. It is shown that the generalized minimal residual (GMRES) method applied to the preconditioned system converges uniformly provided that the coarsest mesh size is reasonably small (but independent of the fine mesh size) and the parameter for the “coarse mesh” space solver is sufficiently large. Numerical experiments show the efficiency of the proposed approach.

关键词： Convergence GMRES method Edge finite element Time-harmonic Maxwell equations

Tilt grain boundaries of hexagonal structures: a spectral viewpoint

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arXiv 2021年

作者： Jiang, Kai Si, Wei Xu, Jie School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan411105 China Chinese Academy of Sciences Beijing China

We propose a spectral viewpoint for grain boundaries that are generally quasiperiodic. To accurately capture the spectra computationally, it is crucial to adopt the projection method for quasiperiodic functions. Armed with the Lifshitz–Petrich free energy, we take the spectral viewpoint to examine tilt grain boundaries of the hexagonal phase. Several ingredients of grain boundaries are extracted, which are not easy to obtain from real-space profiles. We find that only a few spectra substantially contribute to the formation of grain boundaries. Their linear relation to the intrinsic spectra of the bulk hexagonal phase is independent of the tilt angle. By examining the feature of the spectral intensities, we propose a definition of the interface width. The widths calculated from this definition are consistent with visual *** Codes 65N35, 70H12, 82B24 © 2021, CC BY.

关键词： Grain boundaries

Svgd: A Virtual Gradients Descent Method for Stochastic Optimization

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arXiv 2019年

作者： Li, Zheng Shu, Shi School of Mathematics and Computational Science Xiangtan University Xiangtan Hunan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan411105 China

Inspired by dynamic programming, we propose Stochastic Virtual Gradient Descent (SVGD) algorithm where the Virtual Gradient is defined by computational graph and automatic differentiation. The method is computationally efficient and has little memory requirements. We also analyze the theoretical convergence properties and implementation of the algorithm. Experimental results on multiple datasets and network models show that SVGD has advantages over other stochastic optimization methods. Copyright © 2019, The Authors. All rights reserved.

关键词： Machine learning

A high order composite scheme for the second order elliptic problem with nonlocal boundary and its fast algorithm

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Applied Mathematics and computation 2014年 227卷 212-221页

作者： Cunyun Nie Shi Shu Haiyuan Yu Qianjiang An Department of Mathematics and Physics The Hunan Institution of Engineering Xiangtan City Hunan 411104 China Hunan Key Laboratory for Computation & Simulation in Science and Engineering Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education Xiangtan University Hunan 411105 China Department of Mathematics The Institute of Tongren Guizhou 554300 China

The elliptic problem with nonlocal boundary condition is widely applied in the field of science and engineering. Firstly, we construct a linear finite element scheme for the nonlocal boundary problem, and derive the optimal L 2 error estimate. Then, based on the quadratic finite element and the extrapolation linear finite element methods, we present a composite scheme, and prove that it is convergent order three. Furthermore, we design an upper triangular preconditioning algorithm for the linear finite element discrete system. Finally, numerical results not only validate that the new algorithm is efficient, but also show that the new scheme is convergent order three, furthermore order four on uniform grids.

关键词： The elliptic problem with nonlocal boundary condition The finite element method An upper triangular preconditioner Algebraic multigrid method

A NEURAL MULTIGRID SOLVER FOR HELMHOLTZ EQUATIONS WITH HIGH WAVENUMBER AND HETEROGENEOUS MEDIA

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arXiv 2024年

作者： Cui, Chen Jiang, Kai Shu, Shi Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

Solving high-wavenumber and heterogeneous Helmholtz equations presents a longstanding challenge in scientific computing. In this paper, we introduce a deep learning-enhanced multigrid solver to address this issue. By conducting error analysis on standard multigrid applied to a discrete Helmholtz equation, we devise a strategy to handle errors with different frequencies separately. For error components with frequencies distant from the wavenumber, we perform simple smoothing based on local operations at different levels to eliminate them. On the other hand, to address error components with frequencies near the wavenumber, we utilize another multigrid V-cycle to solve an advection-diffusion-reaction (ADR) equation at a coarse scale. The resulting solver, named Wave-ADR-NS, involves parameters learned through unsupervised training. Numerical results demonstrate that Wave-ADR-NS effectively resolves heterogeneous 2D Helmholtz equation with wavenumber up to 2000. Comparative experiments against classical multigrid preconditioners and existing deep learning-based multigrid preconditioners reveals the superior performance of *** Codes 65N22, 65N55, 68T07 Copyright © 2024, The Authors. All rights reserved.

关键词： Helmholtz equation

Convergent analysis of algebraic multigrid method with data-driven parameter learning for non-selfadjoint elliptic problems

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arXiv 2024年

作者： Zhang, Juan Luo, Junyue Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

In this paper, we apply the practical GADI-HS iteration as a smoother in algebraic multigrid (AMG) method for solving second-order non-selfadjoint elliptic problem. Additionally, we prove the convergence of the derived algorithm and introduce a data-driven parameter learing method called Gaussian process regression (GPR) to predict optimal parameters. Numerical experimental results show that using GPR to predict parameters can save a significant amount of time cost and approach the optimal parameters accurately. © 2024, CC BY.

关键词： Algebra