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GEOMETRIC DECOMPOSITION AND EFFICIENT IMPLEMENTATION OF HIGH ORDER FACE AND EDGE ELEMENTS

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arXiv 2023年

作者： Chen, Chunyu Chen, Long Huang, Xuehai Wei, Huayi School of Mathematics and Computational Science Xiangtan University National Center of Applied Mathematics in Hunan Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan411105 China Department of Mathematics University of California at Irvine IrvineCA92697 United States School of Mathematics Shanghai University of Finance and Economics Shanghai200433 China

This study investigates high-order face and edge elements in finite element methods, with a focus on their geometric attributes, indexing management, and practical application. The exposition begins by a geometric decomposition of Lagrange finite elements, setting the foundation for further analysis. The discussion then extends to H(div)-conforming and H(curl)-conforming finite element spaces, adopting variable frames across differing sub-simplices. The imposition of tangential or normal continuity is achieved through the strategic selection of corresponding bases. The paper concludes with a focus on efficient indexing management strategies for degrees of freedom, offering practical guidance to researchers and engineers. It serves as a comprehensive resource that bridges the gap between theory and *** Codes 65N30, 35Q60 Copyright © 2023, The Authors. All rights reserved.

关键词： Degrees of freedom (mechanics)

Fractal and multifractal properties of a family of fractal networks

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Journal of Statistical Mechanics: Theory and Experiment 2014年第2期2014卷

作者： Li, Bao-Gen Yu, Zu-Guo Zhou, Yu Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan HUNAN 411105 China School of Mathematical Sciences Queensland University of Technology Brisbane GPO Box 2434 Q4001 Australia

In this work, we study the fractal and multifractal properties of a family of fractal networks introduced by Gallos et al (2007 Proc. Nat. Acad. Sci. USA 104 7746). In this fractal network model, there is a parameter e which is between 0 and 1, and allows for tuning the level of fractality in the network. Here we examine the multifractal behavior of these networks, the dependence relationship of the fractal dimension and the multifractal parameters on parameter e. First, we find that the empirical fractal dimensions of these networks obtained by our program coincide with the theoretical formula given by Song et al (2006 Nature Phys. 2 275). Then from the shape of the τ(q) and D(q) curves, we find the existence of multifractality in these networks. Last, we find that there exists a linear relationship between the average information dimension D(1) and the parameter e. © 2014 IOP Publishing Ltd and SISSA Medialab srl.

关键词： network dynamics networks random graphs

Determination of multifractal dimensions of complex networks by means of the sandbox algorithm

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Chaos 2015年第2期25卷 023103页

作者： Liu, Jin-Long Yu, Zu-Guo Anh, Vo Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan Hunan 411105 China School of Mathematical Sciences Queensland University of Technology GPO Box 2434 Brisbane Q4001 Australia

Complex networks have attracted much attention in diverse areas of science and technology. Multifractal analysis (MFA) is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. In this paper, we employ the sandbox (SB) algorithm proposed by Tél et al. (Physica A 159, 155-166 (1989)), for MFA of complex networks. First, we compare the SB algorithm with two existing algorithms of MFA for complex networks: the compact-box-burning algorithm proposed by Furuya and Yakubo (Phys. Rev. E 84, 036118 (2011)), and the improved box-counting algorithm proposed by Li et al. (J. Stat. Mech.: Theor. Exp. 2014, P02020 (2014)) by calculating the mass exponents ?(q) of some deterministic model networks. We make a detailed comparison between the numerical and theoretical results of these model networks. The comparison results show that the SB algorithm is the most effective and feasible algorithm to calculate the mass exponents ?(q) and to explore the multifractal behavior of complex networks. Then, we apply the SB algorithm to study the multifractal property of some classic model networks, such as scale-free networks, small-world networks, and random networks. Our results show that multifractality exists in scale-free networks, that of small-world networks is not obvious, and it almost does not exist in random networks. © 2015 AIP Publishing LLC.

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Multifractal temporally weighted detrended cross-correlation analysis to quantify power-law cross-correlation and its application to stock markets

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Chaos 2017年第6期27卷 063111-063111页

作者： Wei, Yun-Lan Yu, Zu-Guo Zou, Hai-Long Anh, Vo Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan 411105 China School of Mathematical Sciences Queensland University of Technology GPO Box 2434 Brisbane Q4001 Australia

A new method-multifractal temporally weighted detrended cross-correlation analysis (MF-TWXDFA)-is proposed to investigate multifractal cross-correlations in this paper. This new method is based on multifractal temporally weighted detrended fluctuation analysis and multifractal cross-correlation analysis (MFCCA). An innovation of the method is applying geographically weighted regression to estimate local trends in the nonstationary time series. We also take into consideration the sign of the fluctuations in computing the corresponding detrended cross-covariance function. To test the performance of the MF-TWXDFA algorithm, we apply it and the MFCCA method on simulated and actual series. Numerical tests on artificially simulated series demonstrate that our method can accurately detect long-range cross-correlations for two simultaneously recorded series. To further show the utility of MF-TWXDFA, we apply it on time series from stock markets and find that power-law cross-correlation between stock returns is significantly multifractal. A new coefficient, MF-TWXDFA cross-correlation coefficient, is also defined to quantify the levels of cross-correlation between two time series. © 2017 Author(s).

关键词： CORRELATION (Statistics) STOCK price indexes REGRESSION analysis TIME series analysis COEFFICIENTS (Statistics)

AN ADAPTIVE FINITE ELEMENT METHOD FOR TWO-DIMENSIONAL ELLIPTIC EQUATIONS WITH LINE DIRAC SOURCES

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arXiv 2021年

作者： Cao, Huihui Li, Hengguang Yi, Nianyu Yin, Peimeng Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Wayne State University Department of Mathematics DetroitMI48202 United States Multiscale Methods and Dynamics Group Computer Science and Mathematics Division Oak Ridge National Laboratory Oak RidgeTN37831 United States

In this paper, we propose a novel adaptive finite element method for an elliptic equation with line Dirac delta functions as a source term. We first study the well-posedness and global regularity of the solution in the whole domain. Instead of regularizing the singular source term and using the classical residual-based a posteriori error estimator, we propose a novel a posteriori estimator based on an equivalent transmission problem with zero source term and nonzero flux jumps on line fractures. The transmission problem is defined in the same domain as the original problem excluding on line fractures, and the solution is therefore shown to be more regular. The estimator relies on meshes conforming to the line fractures and its edge jump residual essentially uses the flux jumps of the transmission problem on line fractures. The error estimator is proven to be both reliable and efficient, an adaptive finite element algorithm is proposed based on the error estimator and the bisection refinement method. Numerical tests show that quasi-optimal convergence rates are achieved even for high order approximations and the adaptive meshes are only locally refined at singular points. © 2021, CC BY-NC-ND.

关键词： Fracture

A Multi-scale Generalized Shrinkage Threshold Network for Image Blind Deblurring in Remote Sensing

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arXiv 2023年

作者： Feng, Yujie Yang, Yin Fan, Xiaohong Zhang, Zhengpeng Zhang, Jianping School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory for Intelligent Computing and Information Processing The Ministry of Education Xiangtan411105 China School of Mathematics and Computational Science Xiangtan University National Center for Applied Mathematics in Hunan Hunan International Scientific and Technological Innovation Cooperation Base of Computational Science Xiangtan411105 China School of Automation and Electronic Information Xiangtan University Xiangtan411105 China

MSC Codes 54H30, 68U10, 94A08Remote sensing images are essential for many applications of the earth’s sciences, but their quality can usually be degraded due to limitations in sensor technology and complex imaging environments. To address this, various remote sensing image deblurring methods have been developed to restore sharp and high-quality images from degraded observational data. However, most traditional model-based deblurring methods usually require predefined hand-crafted prior assumptions, which are difficult to handle in complex applications. On the other hand, deep learning-based deblurring methods are often considered as black boxes, lacking transparency and interpretability. In this work, we propose a new blind deblurring learning framework that utilizes alternating iterations of shrinkage thresholds. This framework involves updating blurring kernels and images, with a theoretical foundation in network design. Additionally, we propose a learnable blur kernel proximal mapping module to improve the accuracy of the blur kernel reconstruction. Furthermore, we propose a deep proximal mapping module in the image domain, which combines a generalized shrinkage threshold with a multi-scale prior feature extraction block. This module also incorporates an attention mechanism to learn adaptively the importance of prior information, improving the flexibility and robustness of prior terms, and avoiding limitations similar to hand-crafted image prior terms. Consequently, we design a novel multi-scale generalized shrinkage threshold network (MGSTNet) that focuses specifically on learning deep geometric prior features to enhance image restoration. Experimental results on real and synthetic remote sensing image datasets demonstrate the superiority of our MGSTNet framework compared to existing deblurring methods. Copyright © 2023, The Authors. All rights reserved.

关键词： Shrinkage

A CONSERVATIVE RELAXATION CRANK-NICOLSON FINITE ELEMENT METHOD FOR THE SCHRÖDINGER-POISSON EQUATION

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arXiv 2024年

作者： Liu, Huini Yi, Nianyu Yin, Peimeng School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China Department of Mathematical Sciences The University of Texas at El Paso El PasoTX79968 United States

In this paper, we propose a novel mass and energy conservative relaxation Crank-Nicolson finite element method for the Schrödinger-Poisson equation. Utilizing only a single auxiliary variable, we simultaneously reformulate the distinct nonlinear terms present in both the Schrödinger equation and the Poisson equation into their equivalent expressions, constructing an equivalent system to the original Schrödinger-Poisson equation. Our proposed scheme, derived from this new system, operates linearly and bypasses the need to solve the nonlinear coupled equation, thus eliminating the requirement for iterative techniques. We in turn rigorously derive error estimates for the proposed scheme, demonstrating second-order accuracy in time and (k+1)th order accuracy in space when employing polynomials of degree up to k. Numerical experiments validate the accuracy and effectiveness of our method and emphasize its conservation properties over long-time *** Codes 35Q55, 65M15, 65M60 © 2024, CC BY-NC-ND.

关键词： Poisson equation

Stability of diverse dodecagonal quasicrystals in T-shaped liquid crystalline molecules

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arXiv 2024年

作者： Wang, Xin Shi, An-Chang Zhang, Pingwen Jiang, Kai Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Department of Physics and Astronomy McMaster University HamiltonONL8S4M1 Canada School of Mathematics and Statistics Wuhan University Wuhan430072 China

Quasicrystals are intriguing ordered structures characterized by the lack of translational symmetry and the existence of rotational symmetry. The tiling of different geometric units such as triangles and squares in two-dimensional space can result in a great variety of quasicrystals that could be realized by the self-assembly of liquid crystalline molecules. In this study, we introduce three self-similar dodecagonal tilings, including a novel Diamond-Square-Triangle pattern, composed of triangular and quadrangular tiles and examine their thermodynamic stability by using the self-consistent field theory applied to T-shaped liquid crystalline molecules. Specifically, we detail the inflation rules for the construction of these dodecagonal tilings and analyze their self-similarity, and show that these tilings can be viewed as projections of higher-dimensional periodic lattice points with projection windows. Using these dodecagonal tilings as initial configurations of the SCFT results in solutions corresponding to quasicrystals that could form from the T-shaped liquid crystalline molecules. The relative stability of these aperiodic phases is analyzed to obtain design rules that could stabilize quasicrystals. Meanwhile, we provide two criteria for distinguishing three dodecagonal quasicrystals and their approximants by analyzing their diffraction peaks. These findings shed new lighten on the discovery of new quasicrystals in soft materials. Copyright © 2024, The Authors. All rights reserved.

关键词： Quasicrystals

A unifying framework for n-dimensional quasi-conformal mappings

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arXiv 2021年

作者： Zhang, Daoping Choi, Gary P.T. Zhang, Jianping Lui, Lok Ming School of Mathematical Sciences Nankai University Tianjin300071 China Department of Mathematics Massachusetts Institute of Technology CambridgeMA02139 United States School of Mathematics and Computational Science Hunan National Center for Applied Mathematics Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China Department of Mathematics The Chinese University of Hong Kong Hong Kong

With the advancement of computer technology, there is a surge of interest in effective mapping methods for objects in higher-dimensional spaces. To establish a one-to-one correspondence between objects, higher-dimensional quasi-conformal theory can be utilized for ensuring the bijectivity of the mappings. In addition, it is often desirable for the mappings to satisfy certain prescribed geometric constraints and possess low distortion in conformality or volume. In this work, we develop a unifying framework for computing n-dimensional quasi-conformal mappings. More specifically, we propose a variational model that integrates quasi-conformal distortion, volumetric distortion, landmark correspondence, intensity mismatch and volume prior information to handle a large variety of deformation problems. We further prove the existence of a minimizer for the proposed model and devise efficient numerical methods to solve the optimization problem. We demonstrate the effectiveness of the proposed framework using various experiments in two- and three-dimensions, with applications to medical image registration, adaptive remeshing and shape modeling. Copyright © 2021, The Authors. All rights reserved.

关键词： Numerical methods