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arXiv 2023年

作者： Dai, Xiaoying Zhou, Aihui Zhou, Yuzhi LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China CAEP Software Center for High Performance Numerical Simulation Beijing100088 China Institute of Applied Physics and Computational Mathematics Beijing100088 China

In this paper, we propose an extended plane wave framework to make the electronic structure calculations of the twisted bilayer 2D material systems practically feasible. Based on the foundation in [Y. Zhou, H. Chen, A. Zhou, J. Comput. Phys. 384, 99 (2019)], following extensions take place: (1) an tensor-producted basis set, which adopts PWs in the incommensurate dimensions, and localized basis in the interlayer dimension, (2) a practical application of a novel cutoff techniques we have recently developed, and (3) a quasi-band structure picture under the small twisted angles and weak interlayer coupling limits. With (1) and (2) now the dimensions of Hamiltonian matrix are reduced by about 2 orders of magnitude compared with the original framework. And (3) enables us to better organize the calculations and understand the results. For numerical examples, we study the electronic structures of the linear bilayer graphene lattice system with the magic twisted angle (∼ 1.05◦). The famous flat bands have been reproduced with their features in quantitative agreement with those from experiments and other theoretical calculations. Moreover, the extended framework has much less computational cost compared to the commensurate cell approximations, and is more extendable compared to the traditional model hamiltonians and tight binding models. Finally this framework can readily accommodate nonlinear models thus will laid the foundations for more effective yet accurate Density Functional Theory (DFT) calculations. © 2023, CC BY.

关键词： Electronic structure

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Localization in the Incommensurate Systems: A Plane Wave Study via Effective Potentials

arXiv

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arXiv 2023年

作者： Wang, Ting Zhou, Yuzhi Zhou, Aihui LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China CAEP Software Center for High Performance Numerical Simulation Beijing100088 China Institute of Applied Physics and Computational Mathematics Beijing100094 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

In this paper, we apply the effective potentials in the localization landscape theory (Filoche et al., 2012, Arnold et al., 2016) to study the spectral properties of the incommensurate systems. We uniquely develop a plane wave method for the effective potentials of the incommensurate systems and utilize that, the localization of the electron density can be inferred from the effective potentials. Moreover, we show that the spectrum distribution can also be obtained from the effective potential version of Weyl’s law. We perform some numerical experiments on some typical incommensurate systems, showing that the effective potential provides an alternative tool for investigating the localization and spectrum distribution of the systems. Copyright © 2023, The Authors. All rights reserved.

关键词： Elastic waves

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ORDER REDUCED METHODS FOR QUAD-CURL EQUATIONS WITH NAVIER TYPE BOUNDARY CONDITIONS

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Journal of computational mathematics 2020年第4期38卷 565-579页

作者： Weifeng Zhang Shuo Zhang Department of Engineering Mechanics and Department of Applied Mathematics Kunming University of Science and TechnologyKunming 650500China Shuo Zhang LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and System SciencesChinese Academy of SciencesBeijing 100190China

Quad-curl equations with Navier type boundary conditions are studied in this *** order reduced formulations equivalent to the model problems are presented,and finite element discretizations are *** convergence rates a... 详细信息

关键词： Quad-curl equation Order reduced scheme Regularity analysis Finite element method

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BOUNDED KRNET AND ITS APPLICATIONS TO DENSITY ESTIMATION AND APPROXIMATION

arXiv

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arXiv 2023年

作者： Zeng, Li Wan, Xiaoliang Zhou, Tao Fuzhou University Fuzhou China Department of Mathematics Center for Computation and Technology Louisiana State University Baton Rouge70803 United States Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

In this paper, we develop an invertible mapping, called B-KRnet, on a bounded domain and apply it to density estimation/approximation for data or the solutions of PDEs such as the Fokker-Planck equation and the Keller-Segel equation. Similar to KRnet, the structure of B-KRnet adapts the pseudo-triangular structure into a normalizing flow model. The main difference between B-KRnet and KRnet is that B-KRnet is defined on a hypercube while KRnet is defined on the whole space, in other words, a new mechanism is introduced in B-KRnet to maintain the exact invertibility. Using B-KRnet as a transport map, we obtain an explicit probability density function (PDF) model that corresponds to the pushforward of a prior (uniform) distribution on the hypercube. It can be directly applied to density estimation when only data are available. By coupling KRnet and B-KRnet, we define a deep generative model on a high-dimensional domain where some dimensions are bounded and other dimensions are unbounded. A typical case is the solution of the stationary kinetic Fokker-Planck equation, which is a PDF of position and momentum. Based on B-KRnet, we develop an adaptive learning approach to approximate partial differential equations whose solutions are PDFs or can be treated as PDFs. A variety of numerical experiments is presented to demonstrate the effectiveness of B-KRnet. Copyright © 2023, The Authors. All rights reserved.

关键词： Fokker Planck equation

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A mathematical Aspect of Bloch's Theorem

arXiv

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arXiv 2024年

作者： Li, Yan Yang, Bin Zhou, Aihui LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China NCMIS Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

In this paper, by studying a class of 1-D Sturm-Liouville problems with periodic coefficients, we show and classify the solutions of periodic Schrodinger equations in a multidimensional case, which tells that not all the solutions are Bloch solutions. In addition, we also provide several properties of the solutions and quasimomenta and illustrate the relationship between bounded solutions and Bloch solutions. Copyright © 2024, The Authors. All rights reserved.

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A predictor-corrector deep learning algorithm for high dimensional stochastic partial differential equations

arXiv

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arXiv 2022年

作者： Zhang, He Zhang, Ran Zhou, Tao School of Mathematics Jilin University Jilin Changchun130012 China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

In this paper, we present a deep learning-based numerical method for approximating high dimensional stochastic partial differential equations (SPDEs). At each time step, our method relies on a predictor-corrector procedure. More precisely, we decompose the original SPDE into a degenerate SPDE and a deterministic PDE. Then in the prediction step, we solve the degenerate SPDE with the Euler scheme, while in the correction step we solve the second-order deterministic PDE by deep neural networks via its equivalent backward stochastic differential equation (BSDE). Under standard assumptions, error estimates and the rate of convergence of the proposed algorithm are presented. The efficiency and accuracy of the proposed algorithm are illustrated by numerical examples. Copyright © 2022, The Authors. All rights reserved.

关键词： Numerical methods

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LOW-DEGREE ROBUST HELLINGER-REISSNER FINITE ELEMENT SCHEMES FOR PLANAR LINEAR ELASTICITY WITH SYMMETRIC STRESS TENSORS

arXiv

引用

arXiv 2022年

作者： Zhang, Shuo Lsec Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences Beijing100049 China

In this paper, we study the construction of low-degree robust finite element schemes for planar linear elasticity on general triangulations. Firstly, we present a low-degree nonconformingHelling-Reissner finite element scheme. For the stress tensor space, the piecewise polynomial shape function space is (Formula presented) the dimension of the total space is asymptotically 8 times of the number of vertices, and the supports of the basis functions are each a patch of an edge. The piecewise rigid body space is used for the displacement. Robust error estimations in L2and broken H(div) norms are presented. Secondly, we investigate the theoretical construction of schemes with lowest-degree polynomial shape function spaces. Specifically, a Hellinger-Reissner finite element scheme is constructed, with the local shape function space for the stress tensor being (Formula presented) which is of the lowest degree for the local approximation of H(div;S), and the space for the displacement is piecewise constants. Robust error estimations in L2and broken H(div) norms are presented for regular solutions and data. Meanwhile, accompanied with this Hellinger-Reissner finite element scheme, a minimal Navier-Lamé finite element scheme is constructed, with the local shape function space for the displacement being (Formula presented) which is of lowest degree for the local approximation in the norm ||·||0+ ||ϵ(·)||0. Robust error estimations in broken energy norms are presented for regular solutions and data. Copyright © 2022, The Authors. All rights reserved.

关键词： Stress tensor

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THE TRUNC ELEMENT IN ANY DIMENSION AND APPLICATION TO A MODIFIED POISSON EQUATION

arXiv

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arXiv 2024年

作者： Li, Hongliang Ming, Pingbing Zhou, Yinghong Department of Mathematics Sichuan Normal University Chengdu610066 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing AMSS Chinese Academy of Sciences No. 55 East Road Zhong-Guan-Cun Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

We introduce a novel TRUNC finite element in n dimensions, encompassing the traditional TRUNC triangle as a particular instance. By establishing the weak continuity identity, we identify it as crucial for error estimate. This element is utilized to approximate a modified Poisson equation defined on a convex polytope, originating from the nonlocal electrostatics model. We have substantiated a uniform error estimate and conducted numerical tests on both the smooth solution and the solution with a sharp boundary layer, which align with the theoretical predictions. Copyright © 2024, The Authors. All rights reserved.

关键词： Electrostatics

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A PRIMAL FINITE ELEMENT SCHEME OF THE HODGE LAPLACE PROBLEM

arXiv

引用

arXiv 2022年

作者： Zhang, Shuo LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences Beijing100049 China

In this paper, a unified family, for any n ≽ 2 and 1 ≼ k ≼ n − 1, of nonconforming finite element schemes are presented for the primal weak formulation of the n-dimensional Hodge-Laplace equation on (Equation presented) and on the simplicial subdivisions of the domain. The finite element scheme possesses an O(h)-order convergence rate for sufficiently regular data, and an O(hs)order rate on any s-regular domain, 0 Copyright © 2022, The Authors. All rights reserved.

关键词： Finite element method

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PARTIALLY ADJOINT DISCRETIZATIONS OF ADJOINT OPERATORS AND NONCONFORMING FINITE ELEMENT EXTERIOR CALCULUS

arXiv

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arXiv 2022年

This paper concerns the discretizations in pair of adjoint operators between Hilbert spaces so that the adjoint properties can be preserved. Due to the finite-dimensional essence of discretized operators, a new framework, theory of partially adjoint operators, is motivated and presented in this paper, so that adjoint properties can be figured out for finite-dimensional operators which can not be non-trivially densely defined in other background spaces. A formal methodology is presented under the framework to construct partially adjoint discretizations by a conforming discretization (CD) and an accompanied-by-conforming discretization (ABCD) for each of the operators. Moreover, the methodology leads to an asymptotic uniformity of an infinite family of finite-dimensional operators. A theory of nonconforming finite element exterior calculus (NCFEEC) is developed by constructing a systematic family of nonconforming finite element differential forms with certain structures by the theoretical framework and the formal construction of discretizations;by the NCFEEC, together with the classical FEEC, adjoint properties are reconstructed between discretizations of adjoint operators and between discretizations of variational *** Codes 47A05, 47A65, 47N40, 65J10, 65N30 Copyright © 2022, The Authors. All rights reserved.

关键词： Calculations

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