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arXiv 2017年

作者： Yue, Xiaoqiang Xu, Yehong Shu, Shi Liu, Menghuan Bu, Weiping School of Mathematics and Computational Science Xiangtan University Hunan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan411105 China

In this study we construct a time-space finite element (FE) scheme and furnish cost-efficient approximations for one-dimensional multi-term time fractional advection diffusion equations on a bounded domain Ω. Firstly, a fully discrete scheme is obtained by the linear FE method in both temporal and spatial directions, and many characterizations on the resulting matrix are established. Secondly, the condition number estimation is proved, an adaptive algebraic multigrid (AMG) method is further developed to lessen computational cost and analyzed in the classical framework. Finally, some numerical experiments are implemented to reach the saturation error order in the L2(Ω) norm sense, and present theoretical confirmations and predictable behaviors of the proposed algorithm.35R11, 65F10, 65F15, 65N55 Copyright © 2017, The Authors. All rights reserved.

关键词： computational complexity

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AN EFFICIENT NUMERICAL METHOD FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH TWO CAPUTO DERIVATIVES

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Journal of computational Mathematics 2016年第2期34卷 113-134页

作者： Shuiping Yang Aiguo Xiao Department of Mathematics Huizhou University Guangdong 516007 China School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan 411105 China

In this paper, we study the Hermite cubic spline collocation method with two parame- ters for solving a initial value problem （IVP） of nonlinear fractional differential equations with two Caputo derivatives. The convergence and nonlinear stability of the method are established. Some illustrative examples are provided to verify our theoretical results. The numerical results also indicate that the convergence order is min{4 - α, 4 - β}, where 0 〈β〈 αa 〈 1 are two parameters associated with the fractional differential equations.

关键词： Fractional differential equations Caputo derivatives Spline collocation method Convergence Stability.

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DLTree: efficient and accurate phylogeny reconstruction using the dynamical language method

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Bioinformatics (Oxford, England) 2017年第14期33卷 2214-2215页

作者： Qi Wu Zu-Guo Yu Jianyi Yang 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 Hunan China. School of Mathematical Sciences Nankai University Tianjin China.

SUMMARY:A number of alignment-free methods have been proposed for phylogeny reconstruction over the past two decades. But there are some long-standing challenges in these methods, including requirement of huge computer memory and CPU time, and existence of duplicate computations. In this article, we address these challenges with the idea of compressed vector, fingerprint and scalable memory management. With these ideas we developed the DLTree algorithm for efficient implementation of the dynamical language model and whole genome-based phylogenetic analysis. The DLTree algorithm was compared with other alignment-free tools, demonstrating that it is more efficient and accurate for phylogeny reconstruction. AVAILABILITY AND IMPLEMENTATION:The DLTree algorithm is freely available at http://***. CONTACT:yuzuguo@*** or yangjy@***. SUPPLEMENTARY INFORMATION:Supplementary data are available at Bioinformatics online.

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An adaptive BDDC algorithm in variational form for mortar discretizations

arXiv

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arXiv 2017年

作者： Peng, Jie Shu, Shi Wang, Junxian School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan411105 China

A balancing domain decomposition by constraints (BDDC) algorithm with adaptive primal constraints in variational form is introduced and analyzed for high-order mortar discretization of two-dimensional elliptic problems with high varying and random coefficients. Some vector-valued auxiliary spaces and operators with essential properties are defined to describe the variational algorithm, and the coarse space is formed by using a transformation operator on each interface. Compared with the adaptive BDDC algorithms for conforming Galerkin approximations, our algorithm is more simple, because there is not any continuity constraints at subdomain vertices in the mortar method involved in this paper. The condition number of the preconditioned system is proved to be bounded above by a user-defined tolerance and a constant which is dependent on the maximum number of interfaces per subdomain, and independent of the mesh size and the contrast of the given coefficients. Numerical results show the robustness and efficiency of the algorithm for various model problems.65N30, 65F10, 65N55 Copyright © 2017, The Authors. All rights reserved.

关键词： Number theory

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Multifractal analysis and topological properties of a new family of weighted Koch networks

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Physica A: Statistical Mechanics and its Applications 2017年 469卷 695-705页

作者： Da-Wen Huang Zu-Guo Yu 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 School of Mathematical Sciences Queensland University of Technology GPO Box 2434 Brisbane Q4001 Australia

Weighted complex networks, especially scale-free networks, which characterize real-life systems better than non-weighted networks, have attracted considerable interest in recent years. Studies on the multifractality of weighted complex networks are still to be undertaken. In this paper, inspired by the concepts of Koch networks and Koch island, we propose a new family of weighted Koch networks, and investigate their multifractal behavior and topological properties. We find some key topological properties of the new networks: their vertex cumulative strength has a power-law distribution; there is a power-law relationship between their topological degree and weight strength; the networks have a high weighted clustering coefficient of 0.41004 (which is independent of the scaling factor c ) in the limit of large generation t ; the second smallest eigenvalue μ 2 and the maximum eigenvalue μ n are approximated by quartic polynomials of the scaling factor c for the general Laplacian operator, while μ 2 is approximately a quartic polynomial of c and μ n = 1.5 for the normalized Laplacian operator. Then, we find that weighted koch networks are both fractal and multifractal, their fractal dimension is influenced by the scaling factor c . We also apply these analyses to six real-world networks, and find that the multifractality in three of them are strong.

关键词： Weighted Koch networks Modified sandbox algorithm Multifractal analysis Topological property

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A novel genome signature based on inter-nucleotide distances profiles for visualization of metagenomic data

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Physica A: Statistical Mechanics and its Applications 2017年 482卷 87-94页

作者： Xian-Hua Xie Zu-Guo Yu Yuan-Lin Ma Guo-Sheng Han 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 Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques Gannan Normal University Jiangxi 341000 China School of Mathematical Sciences Queensland University of Technology GPO Box 2434 Brisbane Q4001 Australia

There has been a growing interest in visualization of metagenomic data. The present study focuses on the visualization of metagenomic data using inter-nucleotide distances profile. We first convert the fragment sequences into inter-nucleotide distances profiles. Then we analyze these profiles by principal component analysis. Finally the principal components are used to obtain the 2-D scattered plot according to their source of species. We name our method as inter-nucleotide distances profiles (INP) method. Our method is evaluated on three benchmark data sets used in previous published papers. Our results demonstrate that the INP method is good, alternative and efficient for visualization of metagenomic data.

关键词： Alignment-free method Inter-nucleotide distance Metagenome Visualization

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Fractal and multifractal analyses of bipartite networks

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Scientific reports 2017年第1期7卷 45588页

作者： Jin-Long Liu Jian Wang Zu-Guo Yu Xian-Hua Xie Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education and 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.

Bipartite networks have attracted considerable interest in various fields. Fractality and multifractality of unipartite (classical) networks have been studied in recent years, but there is no work to study these properties of bipartite networks. In this paper, we try to unfold the self-similarity structure of bipartite networks by performing the fractal and multifractal analyses for a variety of real-world bipartite network data sets and models. First, we find the fractality in some bipartite networks, including the CiteULike, Netflix, MovieLens (ml-20m), Delicious data sets and (u, v)-flower model. Meanwhile, we observe the shifted power-law or exponential behavior in other several networks. We then focus on the multifractal properties of bipartite networks. Our results indicate that the multifractality exists in those bipartite networks possessing fractality. To capture the inherent attribute of bipartite network with two types different nodes, we give the different weights for the nodes of different classes, and show the existence of multifractality in these node-weighted bipartite networks. In addition, for the data sets with ratings, we modify the two existing algorithms for fractal and multifractal analyses of edge-weighted unipartite networks to study the self-similarity of the corresponding edge-weighted bipartite networks. The results show that our modified algorithms are feasible and can effectively uncover the self-similarity structure of these edge-weighted bipartite networks and their corresponding node-weighted versions.

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Exact and numerical stability analysis of reaction-diffusion equations with distributed delays

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Frontiers of Mathematics in China 2016年第1期11卷 189-205页

作者： Gengen ZHANG Aiguo XIAO School of Mathematics and Computational Science Hunan Key Laboratory for Computationand Simulation in Science and Engineering Xiangtan University Xiangtan 411105 China

This paper is concerned with the stability analysis of the exact and numerical solutions of the reaction-diffusion equations with distributed delays. This kind of partial integro-differential equations contains time memory term and delay parameter in the reaction term. Asymptotic stability and dissipativity of the equations with respect to perturbations of the initial condition are obtained. Moreover, the fully discrete approximation of the equations is given. We prove that the one-leg θ-method preserves stability and dissipativity of the underlying equations. Numerical example verifies the efficiency of the obtained method and the validity of the theoretical results.

关键词： keywords Reaction-diffusion equations distributed delay dissipativity,asymptotic stability

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Iteratively linearized reweighted alternating direction method of multipliers for a class of nonconvex problems

arXiv

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arXiv 2017年

作者： Sun, Tao Jiang, Hao Cheng, Lizhi Zhu, Wei Department of Mathematics National University of Defense Technology Changsha Hunan410073 China College of Computer National University of Defense Technology Changsha Hunan410073 China The State Key Laboratory for High Performance Computation National University of Defense Technology Changsha Hunan410073 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan Hunan411105 China

In this paper, we consider solving a class of nonconvex and nonsmooth problems frequently appearing in signal processing and machine learning research. The traditional alternating direction method of multipliers encounters troubles in both mathematics and computations in solving the nonconvex and nonsmooth subproblem. In view of this, we propose a reweighted alternating direction method of multipliers. In this algorithm, all subproblems are convex and easy to solve. We also provide several guarantees for the convergence and prove that the algorithm globally converges to a critical point of an auxiliary function with the help of the Kurdyka-Lojasiewicz property. Several numerical results are presented to demonstrate the efficiency of the proposed algorithm. Copyright © 2017, The Authors. All rights reserved.

关键词： Iterative methods

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A convergence framework for inexact nonconvex and nonsmooth algorithms and its applications to several iterations

arXiv

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arXiv 2017年

作者： Sun, Tao Jiang, Hao Cheng, Lizhi Zhu, Wei Department of Mathematics National University of Defense Technology Changsha Hunan410073 China College of Computer National University of Defense Technology Changsha Hunan410073 China State Key Laboratory for High Performance Computation National University of Defense Technology Changsha Hunan410073 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan Hunan411105 China

In this paper, we consider the convergence of an abstract inexact nonconvex and nonsmooth algorithm, which cannot be included in previous framework. We promise a pseudo sufficient descent condition and a pseudo relative error condition, which are both related to an auxiliary sequence, for the algorithm;and a continuity condition is assumed to hold. In fact, a lot of classical inexact nonconvex and nonsmooth algorithms allow these three conditions. Under an assumption on the auxiliary sequence, we prove the sequence generated by the general algorithm converges to a critical point of the objective function if being assumed semi-Algebraic property. The core of the proofs lies in building a new Lyapunov function, whose successive difference provides a bound for the successive difference of the points generated by the algorithm. And then, we apply our findings to several classical inexact nonconvex iterative algorithms and derive the corresponding convergence results. Copyright © 2017, The Authors. All rights reserved.

关键词： Lyapunov functions

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