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A New Triangular Spectral Element Method II: Mixed Formulation and hp-Error Estimates

Numerical mathematics(Theory,Methods and Applications) 2019年第1期12卷 72-97页

作者： Bingzhen Zhou Bo Wang Li-Lian Wang Ziqing Xie Key Laboratory of High Performance Computing and Stochastic Information Processing(HPCSIP) College of Mathematics and StatisticsHunan Normal UniversityChangsha 410081P.R.China Division of Mathematical Sciences School of Physical and Mathematical SciencesNanyang Technological University637371Singapore

Mixed triangular spectral element method using nodal basis on unstructured meshes is investigated in this *** method is based on equivalent first order system of the elliptic problem and rectangle-triangle *** fully enjoys the ten-sorial structure and flexibility in handling complex domains by using nodal basis and unstructured triangular *** from the usual Galerkin formulation,the mixed form is particularly advantageous in this context,since it can avoid the singularity in-duced by the rectangle-triangle transform in the calculation of the matrices,and does not require the evaluation of the stiffness *** hp a priori error estimate is pres-ented for the proposed *** implementation details and some numerical exam-ples are provided to validate the accuracy and flexibility of the method.

关键词： Triangular spectral element method hp error analysis mixed form interpolation error in H^(1)-norm

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On the Eigenvalues of General Sum-Connectivity Laplacian Matrix

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Journal of the Operations Research Society of China 2013年第3期1卷 347-358页

作者： Hanyuan Deng He Huang Jie Zhang College of Mathematics and Computer Science Key Laboratory of High Performance Computing and Stochastic Information Processing(HPCSIP)(Ministry of Education of China)Hunan Normal UniversityChangshaHunan 410081China Computer Science Department Aarhus UniversityAarhusDenmark

The connectivity index was introduced by Randi´c(***.97(23):6609–6615,1975)and was generalized by Bollobás and Erdös(Ars Comb.50:225–233,1998).It studies the branching property of graphs,and has been applied to studying network *** this paper we focus on the general sum-connectivity index which is a variant of the connectivity *** characterize the tight upper and lower bounds of the largest eigenvalue of the general sum-connectivity matrix,as well as its spectral *** show the corresponding extremal *** addition,we show that the general sum-connectivity index is determined by the eigenvalues of the general sum-connectivity Laplacian matrix.

关键词： Connectivity index Eigenvalue Laplacian matrix

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A Compact Difference Scheme for an Evolution Equation with a Weakly Singular Kernel

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Numerical mathematics(Theory,Methods and Applications) 2012年第4期5卷 559-572页

作者： Hongbin Chen Da Xu Institute of Mathematics and Physics College of ScienceCentral South University of Forestry and TechnologyChangsha410004HunanChina Key Laboratory of High Performance Computing and Stochastic Information Processing(Ministry of Education of China) College of Mathematics and Computer ScienceHunan Normal UniversityChangsha410081HunanChina

This paper is concerned with a compact difference scheme with the truncation error of order 3/2 for time and order 4 for space to an evolution equation with a weakly singular *** integral term is treated by means of the second order convolution quadrature suggested by *** stability and convergence are proved by the energy method.A numerical experiment is reported to verify the theoretical predictions.

关键词： Evolution equation weakly singular kernel compact difference scheme stability convergence numerical experiment

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A Multi-task learning model with low-level feature sharing and inter-feature guidance for segmentation and classification of medical images

A Multi-task learning model with low-level feature sharing a...

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2024 IEEE International Conference on Bioinformatics and Biomedicine, BIBM 2024

作者： Zhang, Rong Xie, Minzhu Hunan Normal University College of Information Science and Engineering Changsha China Hunan Normal University College of Mathematics and Statistics Changsha China Ministry of Education Key Laboratory of Computing and Stochastic Mathematics China

ISBN: (纸本)9798350386226

Medical image segmentation and classification are both crucial components of computer-aided diagnosis, and past studies have identified the inherent correlations between them in various cases. Numerous multi-task models have been developed, with most leveraging shared features extracted by a feature extractor to address both tasks. However, few have paid attention to the feature differences between them which may be obvious when the segmentation object area is larger than the classification object area. To address the issue, we introduce a model named SCMTL-LSFG, which employs a low-level feature sharing and high-level inter-feature guidance strategy. SCMTL-LSFG comprises a segmentation branch and a classification branch, which share a low-level feature extraction component but have two separate high-level feature extraction components. Since there are correlations between the two tasks, SCMTL-LSFG leverages the segmentation component to guide the classification component in high-level feature extraction by the Inter-Feature Guidance module we design. The evaluation conducted on a public breast ultrasound image dataset and a COVID-19 chest X-ray image dataset indicates that SCMTL-LSFG effectively improves classification accuracy. Our experiment results also demonstrated that SCMTL-LSFG significantly outperforms three state-of-art similar models in both tasks. © 2024 IEEE.

关键词： Image segmentation

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Variable 2-Microlocal Besov–Triebel–Lizorkin-Type Spaces

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Acta Mathematica Sinica,English Series 2018年第4期34卷 699-748页

作者： Su Qing WU Da Chun YANG Wen YUAN Ci Qiang ZHUO Laboratory of Mathematics and Complex Systerns (Ministry of Education) School of Mathematical Sciences Beijing Normal University Beijing 100875 P. R. China Key Laboratory of High Performance Computing and Stochastic Information Processing ( HPCSIP) (Ministry of Education of China) College of Mathematics and Computer Science Hu'nan Normal University Changsha 410081 P. R. China

This article is devoted to the study of variable 2-microlocal Besov-type and Triebel- Lizorkin-type spaces. These variable function spaces are defined via a Fourier-analytical approach. The authors then characterize these spaces by means of Q-transforms, Peetre maximal functions, smooth atoms, ball means of differences and approximations by analytic functions. As applications, some re- lated Sobolev-type embeddings and trace theorems of these spaces are Mso established. Moreover, some obtained results, such as characterizations via approximations by analytic functions, are new even for the classical variable Besov and Triebel-Lizorkin spaces.

关键词： 2-Microlocal Besov space 2-Microlocal Triebel-Lizorkin space variable exponent φ-transform Peetre maximal function difference atom embedding

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An Overview of Kernelization Algorithms for Graph Modification Problems

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Tsinghua Science and Technology 2014年第4期19卷 346-357页

作者： Yunlong Liu Jianxin Wang Jiong Guo School of Information Science and Engineering Central South University College of Mathematics and Computer Science Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China) Hunan Normal University Universitt des Saarlandes Campus E 1.7D-66123 Saarbrcken Germany

Kernelization algorithms for graph modification problems are important ingredients in parameterized computation theory. In this paper, we survey the kernelization algorithms for four types of graph modification problems, which include vertex deletion problems, edge editing problems, edge deletion problems, and edge completion problems. For each type of problem, we outline typical examples together with recent results, analyze the main techniques, and provide some suggestions for future research in this field.

关键词： graph modification problem fixed-parameter tractable kernelization algorithm

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THE ULTRACONVERGENCE OF EIGENVALUES FOR BI-QUADRATIC FINITE ELEMENTS

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Journal of Computational mathematics 2012年第5期30卷 555-564页

作者： Lingxiong Meng Zhimin Zhang Key Laboratory of High Performance Computing and Stochastic Information Processing(HPCSIP)(Ministry of Education of China) College of Mathematics and Computer ScienceHunan Normal UniversityChangsha 410081China Department of Mathematics Wayne State UniversityDetroitMI 48202USA and Guangdong Province Key Laboratory of Computational ScienceSchool of Mathematics and Computational ScienceSun Yat-sen UniversityGuangzhou 510275China

The classical eigenvalue problem of the second-order elliptic operator is approxlmateo with hi-quadratic finite element in this paper. We construct a new superconvergent function recovery operator, from which the O（h^8｜ in h｜^2） ultraconvergence of eigenvalue approximation is obtained. Numerical experiments verify the theoretical results.

关键词： Finite element method Eigenvalue recovery Ultraconvergence

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Spectral Petrov-Galerkin Methods for the Second Kind Volterra Type Integro-Differential Equations

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Numerical mathematics(Theory,Methods and Applications) 2011年第2期4卷 216-236页

作者： Xia Tao Ziqing Xie Xiaojun Zhou Key Laboratory of High Performance Computing and Stochastic Information Processing Ministry of Education of ChinaHunan Normal UniversityChangshaHunan 410081P.R.China School of Mathematics and Computer Science Guizhou Normal UniversityGuiyangGuizhou 550001P.R.China

This work is to provide general spectral and pseudo-spectral Jacobi-Petrov-Galerkin approaches for the second kind Volterra integro-differential *** Gauss-Legendre quadrature formula is used to approximate the integral operator and the inner product based on the Jacobi weight is implemented in the weak formulation in the numerical *** some spectral and pseudo-spectral Jacobi-Petrov-Galerkin methods,a rigorous error analysis in both L2_(ω^(α,β))^(2),and L^(∞)norms is given provided that both the kernel function and the source function are sufficiently *** experiments validate the theoretical prediction.

关键词： Volterra integro-differential equation spectral Jacobi-Petrov-Galerkin pseudo-spectral Jacobi-Petrov-Galerkin spectral convergence

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An Extrapolation Cascadic MultigridMethod for Elliptic Problems on Reentrant Domains

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Advances in Applied mathematics and Mechanics 2017年第6期9卷 1347-1363页

作者： Kejia Pan Dongdong He Chuanmiao Chen School of Mathematics and Statistics Central South UniversityChangshaHunan 410083China School of Aerospace Engineering and Applied Mechanics Tongji UniversityShanghai 200092China College of Mathematics and Computer Science Key Laboratory of High Performance Computing and Stochastic Information Processing(Ministry of Education of China)Hunan Normal UniversityChangshaHunan 410081China

This paper proposes an extrapolation cascadic multigrid(EXCMG)method to solve elliptic problems in domains with reentrant *** a class ofλ-graded meshes,we derive some new extrapolation formulas to construct a high-order approximation to the finite element solution on the next finer mesh using the numerical solutions on two-level of grids(current and previous grids).Then,this high-order approximation is used as the initial guess to reduce computational cost of the conjugate gradient *** application of this idea results in the EXCMG method proposed in this ***,numerical results for a crack problem and an L-shaped problem are presented to verify the efficiency and effectiveness of the proposed EXCMG method.

关键词： Richardson extrapolation Cascadicmultigrid gradedmesh elliptic problems corner singularity

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Graph Fourier Transforms on Signed Graphs 24

Graph Fourier Transforms on Signed Graphs

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12th International Conference on Communications and Broadband Networking, ICCBN 2024

作者： Chen, Yang Wang, Menglin Li, Changhao Cheng, Cheng Key Laboratory of Computing and Stochastic Mathematics Ministry of Education School of Mathematics and Statistics Hunan Normal University China School of Mathematics and Statistics Hunan Normal University China Department of Information Science and Technology Jinan University China School of Mathematics Sun Yat-sen University China

ISBN: (纸本)9798400717109

The graph Fourier transform (GFT), is an important tool to identify the patterns and quantify the influence of members and communities of a social network. In this paper, we study the GFT on signed graphs. We decompose signed graphs into positive and negative parts as unsigned subgraphs and construct the graph dilation matrix which is positive semi-definite. Then we propose the GFT on signed graph by the eigen-decomposition of the graph dilation matrix, for which Parseval's identity still holds. We show that bandlimiting in the low frequencies of the proposed GFT provides good approximations to graph signal on signed graphs. Additionally, our method effectively denoises graph signals on signed circulant graphs. © 2024 ACM.

关键词： Graph algorithms

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