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

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

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Multifractal analysis of weighted networks by a modified sandbox algorithm

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Scientific reports 2015年第1期5卷 17628页

作者： Yu-Qin Song Jin-Long Liu Zu-Guo Yu Bao-Gen Li Hunan Key Laboratory for Computation and Simulation in Science and Engineering and Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan Hunan 411105 China. College of Science Hunan University of technology Zhuzhou Hunan 412007 China. School of Mathematical Sciences Queensland University of Technology Brisbane Q4001 Australia.

Complex networks have attracted growing attention in many fields. As a generalization of fractal analysis, multifractal analysis (MFA) is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. Some algorithms for MFA of unweighted complex networks have been proposed in the past a few years, including the sandbox (SB) algorithm recently employed by our group. In this paper, a modified SB algorithm (we call it SBw algorithm) is proposed for MFA of weighted networks. First, we use the SBw algorithm to study the multifractal property of two families of weighted fractal networks (WFNs): "Sierpinski" WFNs and "Cantor dust" WFNs. We also discuss how the fractal dimension and generalized fractal dimensions change with the edge-weights of the WFN. From the comparison between the theoretical and numerical fractal dimensions of these networks, we can find that the proposed SBw algorithm is efficient and feasible for MFA of weighted networks. Then, we apply the SBw algorithm to study multifractal properties of some real weighted networks - collaboration networks. It is found that the multifractality exists in these weighted networks, and is affected by their edge-weights.

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Adaptive mesh refinement and superconvergence for two-dimensional interface problems

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SIAM Journal on Scientific Computing 2014年第4期36卷 A1478-A1499页

作者： Wei, Huayi Chen, Long Huang, Yunqing Zheng, Bin Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Lab. of Intelligent Comp. and Info. Proc. Comp. Min. of Educ. School of Math. and Computational Sci. Xiangtan University Xiangtan Hunan 411105 China Department of Mathematics University of California at Irvine Irvine 92697 CA United States Northwest National Lab Richland 99352 WA United States

Adaptive mesh refinement and the B&#2.6;rgers algorithm are combined to generate a body-fitted mesh which can resolve the interface with fine geometric details. Standard linear finite element method based on such body... 详细信息

Adaptive mesh refinement and the Börgers algorithm are combined to generate a body-fitted mesh which can resolve the interface with fine geometric details. Standard linear finite element method based on such body-fitted meshes is applied to the elliptic interface problem and proven to be superclose to the linear interpolation of the exact solution. Based on this superconvergence result, a maximal norm error estimate of order one and half is obtained without using the discrete maximum principle. The data structure and meshing algorithms, including local refinement and coarsening, are very simple. In particular, no tree structure is needed. An efficient solver for solving the resulting linear algebraic systems is also developed and shown be robust with respect to both the problem size and the jump of the diffusion coefficients. © 2.14 Society for Industrial and Applied Mathematics.

关键词： Adaptive mesh refinement Body-fitted mesh Superconvergence

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Laplacian normalization and random walk on heterogeneous networks for disease-gene prioritization

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computational biology and chemistry 2015年 57卷 21-8页

作者： Zhi-Qin Zhao Guo-Sheng Han Zu-Guo Yu Jinyan Li Hunan Key Laboratory for Computation and Simulation in Science and Engineering and Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan Hunan 411105 China. Hunan Key Laboratory for Computation and Simulation in Science and Engineering and 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. Electronic address: yuzg1970@***. Advanced Analytics Institute & Centre for Health Technologies University of Technology Sydney Broadway NSW 2007 Australia. Electronic address: jinyan.li@uts.edu.au.

Random walk on heterogeneous networks is a recently emerging approach to effective disease gene prioritization. Laplacian normalization is a technique capable of normalizing the weight of edges in a network. We use this technique to normalize the gene matrix and the phenotype matrix before the construction of the heterogeneous network, and also use this idea to define the transition matrices of the heterogeneous network. Our method has remarkably better performance than the existing methods for recovering known gene-phenotype relationships. The Shannon information entropy of the distribution of the transition probabilities in our networks is found to be smaller than the networks constructed by the existing methods, implying that a higher number of top-ranked genes can be verified as disease genes. In fact, the most probable gene-phenotype relationships ranked within top 3 or top 5 in our gene lists can be confirmed by the OMIM database for many cases. Our algorithms have shown remarkably superior performance over the state-of-the-art algorithms for recovering gene-phenotype relationships. All Matlab codes can be available upon email request.

关键词： Disease genes and phenotypes Heterogeneous network Laplacian normalization Leave-one-out cross-validation Random walk with restart Shannon information entropy

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Adaptive tetrahedral mesh generation by constrained centroidal voronoi-delaunay tessellations for finite element methods

Adaptive tetrahedral mesh generation by constrained centroid...

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作者： Chen, Jie Huang, Yunqing Wang, Desheng Xie, Xiaoping School of Mathematics and Statistics Xi'An Jiaotong University Xi'an 710049 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan 411005 China Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University Singapore 637616 Singapore School of Mathematics Sichuan University Chengdu 610064 China

This article presents a tetrahedral mesh adaptivity algorithm for three-dimensional elliptic partial differential equations (PDEs) using finite element methods. The main issues involved are the mesh size and mesh quality, which have great influence on the accuracy of the numerical solution and computational cost. The first issue is addressed by a posteriori error estimator based on superconvergent gradient recovery. The second issue is solved by constrained centroidal Voronoi-Delaunay tessellations (CCVDT), which guarantees good quality tetrahedrons over a large class of mesh domains even, if the grid size varies a lot at any particular refinement level. The CCVDT enjoys the energy equidistribution property so that the errors are very well equidistributed with properly chosen sizing field (density function). And with this good property, a new refinement criteria is raised which is different from the traditional bisection refinement. © 2.13 Wiley Periodicals, Inc.

关键词： Finite element method

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Topological properties and fractal analysis of a recurrence network constructed from fractional Brownian motions

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Physical Review E 2014年第3期89卷 032814-032814页

作者： Jin-Long Liu Zu-Guo Yu Vo Anh Hunan Key Laboratory for Computation and Simulation in Science and Engineering and 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

Many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks. In this work, we investigate the fundamental topological and fractal properties of recurrence networks constructed from fractional Brownian motions (FBMs). First, our results indicate that the constructed recurrence networks have exponential degree distributions; the average degree exponent 〈λ〉 increases first and then decreases with the increase of Hurst index H of the associated FBMs; the relationship between H and 〈λ〉 can be represented by a cubic polynomial function. We next focus on the motif rank distribution of recurrence networks, so that we can better understand networks at the local structure level. We find the interesting superfamily phenomenon, i.e., the recurrence networks with the same motif rank pattern being grouped into two superfamilies. Last, we numerically analyze the fractal and multifractal properties of recurrence networks. We find that the average fractal dimension 〈dB〉 of recurrence networks decreases with the Hurst index H of the associated FBMs, and their dependence approximately satisfies the linear formula 〈dB〉≈2.H, which means that the fractal dimension of the associated recurrence network is close to that of the graph of the FBM. Moreover, our numerical results of multifractal analysis show that the multifractality exists in these recurrence networks, and the multifractality of these networks becomes stronger at first and then weaker when the Hurst index of the associated time series becomes larger from 0.4 to 0.95. In particular, the recurrence network with the Hurst index H=0.5 possesses the strongest multifractality. In addition, the dependence relationships of the average information dimension 〈D(1)〉 and the average correlation dimension 〈D(2.〉 on the Hurst index H can also be fitted well with linear functions. Our results strongly suggest that the recurrence network inherits the basic characteristic

关键词： TOPOLOGICAL property FRACTAL analysis BROWNIAN motion CORRELATION (Statistics) RECURSIVE sequences (Mathematics) COMPUTER networks

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Finite difference and sinc-collocation approximations to a class of fractional diffusion-wave equations

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Journal of Applied Mathematics 2014年第none期2014卷 1-11页

作者： Mao, Zhi Xiao, Aiguo Yu, Zuguo Shi, Long Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan 411105 China Mathematics and Information Engineering Department Tongren University Tongren Guizhou 554300 China

We propose an efficient numerical method for a class of fractional diffusion-wave equations with the Caputo fractional derivative of order α. This approach is based on the finite difference in time and the global sinc collocation in space. By utilizing the collocation technique and some properties of the sinc functions, the problem is reduced to the solution of a system of linear algebraic equations at each time step. Stability and convergence of the proposed method are rigorously analyzed. The numerical solution is of 3 - α order accuracy in time and exponential rate of convergence in space. Numerical experiments demonstrate the validity of the obtained method and support the obtained theoretical results. © 2.14 Zhi Mao et al.

关键词： FINITE difference method COLLOCATION methods APPROXIMATION theory WAVE equation FRACTIONAL calculus CAPUTO fractional derivatives STOCHASTIC convergence

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Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative

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Journal of computational Physics 2013年 242卷 670-681页

作者： Wang, Dongling Xiao, Aiguo Yang, Wei 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

In this paper, the Crank-Nicolson (CN) difference scheme for the coupled nonlinear Schr&#2.6;dinger equations with the Riesz space fractional derivative is studied. The existence of this difference solution is proved ... 详细信息

In this paper, the Crank-Nicolson (CN) difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative is studied. The existence of this difference solution is proved by the Brouwer fixed point theorem. The stability and convergence of the CN scheme are discussed in the L2.norm. When the fractional order is two, all those results are in accord with the difference scheme developed for the classical non-fractional coupled nonlinear Schrödinger equations. Some numerical examples are also presented. © 2.13 Elsevier Inc.

关键词： Nonlinear equations

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Fractal and complex network analyses of protein molecular dynamics

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Physica A: Statistical Mechanics and its Applications 2014年 416卷 21-32页

作者： Yuan-Wu Zhou Jin-Long Liu Zu-Guo Yu Zhi-Qin Zhao Vo Anh Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan 411105 China Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan Hunan 411105 China Civil Construction Engineering Department Guangxi University of Science and Technology Liuzhou 545006 China School of Mathematical Sciences Queensland University of Technology GPO Box 2434 Brisbane Q4001 Australia

Based on protein molecular dynamics, we investigate the fractal properties of energy, pressure and volume time series using the multifractal detrended fluctuation analysis (MF-DFA) and the topological and fractal properties of their converted horizontal visibility graphs (HVGs). The energy parameters of protein dynamics we considered are bonded potential, angle potential, dihedral potential, improper potential, kinetic energy, Van der Waals potential, electrostatic potential, total energy and potential energy. The shape of the h ( q ) curves from MF-DFA indicates that these time series are multifractal. The numerical values of the exponent h ( 2.) of MF-DFA show that the series of total energy and potential energy are non-stationary and anti-persistent; the other time series are stationary and persistent apart from series of pressure (with H ≈ 0.5 indicating the absence of long-range correlation). The degree distributions of their converted HVGs show that these networks are exponential. The results of fractal analysis show that fractality exists in these converted HVGs. For each energy, pressure or volume parameter, it is found that the values of h ( 2.) of MF-DFA on the time series, exponent λ of the exponential degree distribution and fractal dimension d B of their converted HVGs do not change much for different proteins (indicating some universality). We also found that after taking average over all proteins, there is a linear relationship between 〈 h ( 2.) 〉 (from MF-DFA on time series) and 〈 d B 〉 of the converted HVGs for different energy, pressure and volume.

关键词： Protein molecular dynamics Multifractal detrended fluctuation analysis Horizontal visibility graph Fractal analysis Degree distribution

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Sinc-Chebyshev collocation method for a class of fractional diffusion-wave equations

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The Scientific World Journal 2014年第1期2014卷 143983-143983页

作者： Mao, Zhi Xiao, Aiguo Yu, Zuguo Shi, Long 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 Mathematics and Information Engineering Department Tongren University Tongren Guizhou 554300 China

This paper is devoted to investigating the numerical solution for a class of fractional diffusion-wave equations with a variable coefficient where the fractional derivatives are described in the Caputo sense. The approach is based on the collocation technique where the shifted Chebyshev polynomials in time and the sinc functions in space are utilized, respectively. The problem is reduced to the solution of a system of linear algebraic equations. Through the numerical example, the procedure is tested and the efficiency of the proposed method is confirmed. © 2.14 Zhi Mao et al.

关键词： Algorithms Computer simulation Diffusion Models, Chemical Models, Theoretical Numerical Analysis, Computer-Assisted Rheology/methods

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