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A MULTIVARIATE MULTIQUADRIC QUASI-INTERPOLATION WITH QUADRIC REPRODUCTION

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Journal of Computational Mathematics 2012年第3期30卷 311-323页

作者： renzhong feng Xun Zhou School of Mathematics and Systematic Science Beijing University of Aeronautics and AstronauticsBeijing 100191China Key Laboratory of Mathematics Informatics and Behavioral SemanticsMinistry of EducationChina

In this paper, by using multivariate divided differences to approximate the partial derivative and superposition, we extend the multivariate quasi-interpolation scheme based on dimension-splitting technique which can reproduce linear polynomials to the scheme quadric polynomials. Furthermore, we give the approximation error of the modified scheme. Our multivariate multiquadric quasi-interpolation scheme only requires information of lo- cation points but not that of the derivatives of approximated function. Finally, numerical experiments demonstrate that the approximation rate of our scheme is significantly im- proved which is consistent with the theoretical results.

关键词： Quasi-interpolation Multiquadric functions Polynomial reproduction :Pn-exact A-discretization of :Da Approximation error.

Rational Quasi-Interpolation Approximation of Scattered Data in R^(3)

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Numerical Mathematics(Theory,Methods and Applications) 2018年第1期11卷 169-186页

作者： renzhong feng Lifang Song School of Mathematics and Systematic Science&Key Laboratory of Mathematics Informatics and Behavioral SemanticsMinistry of EducationBeijing University of Aeronautics and AstronauticsBeijing 100191China

This paper is concerned with a piecewise smooth rational quasi-interpolation with algebraic accuracy of degree(n+1)to approximate the scattered data in R *** firstly use the modified Taylor expansion to expand the mean value coordinates interpolation with algebraic accuracy of degree one to one with algebraic accuracy of degree(n+1).Then,based on the triangulation of the scattered nodes in R^(2),on each triangle a rational quasi-interpolation function is *** constructed rational quasi-interpolation is a linear combination of three different expanded mean value coordinates interpolations and it has algebraic accuracy of degree(n+1).By comparing accuracy,stability,and efficiency with the C^(1)-Tri-interpolation method of Goodman[16]and the MQ Shepard method,it is observed that our method has some computational advantages.

关键词： Scattered data mean value coordinates interpolation modified Taylor expansion rational quasi-interpolation algebraic accuracy

The Recovery Guarantee for Orthogonal Matching Pursuit Method to Reconstruct Sparse Polynomials

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Numerical Mathematics(Theory,Methods and Applications) 2022年第3期15卷 793-818页

作者： Aitong Huang renzhong feng Sanpeng Zheng School of Mathematics Science&Key Laboratory of Mathematics Informatics and Behavioral SemanticsMinistry of EducationBeihang UniversityBeijingChina

Orthogonal matching pursuit(OMP for short)algorithm is a popular method of sparse signal recovery in compressed *** paper applies OMP to the sparse polynomial reconstruction *** from classical research methods using mutual coherence or restricted isometry property of the measurement matrix,the recovery guarantee and the success probability of OMP are obtained directly by the greedy selection ratio and the probability *** results show that the failure probability of OMP given in this paper is exponential small with respect to the number of sampling *** addition,the recovery guarantee of OMP obtained through classical methods is lager than that of ℓ_(1)-minimization whatever the sparsity of sparse polynomials is,while the recovery guarantee given in this paper is roughly the same as that of ℓ_(1)-minimization when the sparsity is less than ***,the numerical experiments verify the availability of the theoretical results.

关键词： Reconstruction of sparse polynomial uniformly bounded orthogonal system orthogonal matching pursuit method probability of successful reconstruction sub-Gaussian random variable

RECONSTRUCTION OF SPARSE POLYNOMIALS VIA QUASI-ORTHOGONAL MATCHING PURSUIT METHOD

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Journal of Computational Mathematics 2023年第1期41卷 18-38页

作者： renzhong feng Aitong Huang Ming-Jun Lai Zhaiming Shen School of Mathematics Science&Key Laboratory of Mathematics Informatics and Behavioral SemanticsMinistry of EducationBeihang UniversityBeijing 100191China Department of Mathematics University of GeorgiaAthensGA 30602U.S.A

In this paper,we propose a Quasi-Orthogonal Matching Pursuit(QOMP)algorithm for constructing a sparse approximation of functions in terms of expansion by orthonormal *** the two kinds of sampled data,data with noises and without noises,we apply the mutual coherence of measurement matrix to establish the convergence of the QOMP algorithm which can reconstruct s-sparse Legendre polynomials,Chebyshev polynomials and trigonometric polynomials in s step *** results are also extended to general bounded orthogonal system including tensor product of these three univariate orthogonal ***,numerical experiments will be presented to verify the effectiveness of the QOMP method.

关键词： Reconstruction of sparse polynomial Compressive sensing Mutual coherence Quasi-orthogonal matching pursuit algorithm

WEIGHTED MEAN CONVERGENCE OF HAKOPIAN INTERPOLATION ON THE DISK

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Analysis in Theory and Applications 2007年第3期23卷 213-227页

作者： Xuezhang Liang renzhong feng Xuenan Sun Jilin University China Beijing University of Aeronautics and Astronautics China Northeast Normal University China

In this paper, we study weighted mean integral convergence of Hakopian interpolation on the unit disk D. We show that the inner product between Hakopian interpolation polynomial Hn（f;x,y） and a smooth function g（x,y） on D converges to that of f（x,y） and g（x,y） on D when n →∞ , provided f（x,y） belongs to C（D） and all first partial derivatives of g（x,y） belong to the space LipM^α（0 〈 α ≤1）. We further show that provided all second partial derivatives of g（x,y） also belong to the space LipM^α and f（x,y） belongs to C^1 （D）, the inner product between the partial derivative of Hakopian interpolation polynomial δ/δx Hn（f;z,y） and g（x,y） on D converges to that between δ/δxf（x,y） and g（x,y） on D when n →∞. oo.

关键词： Hakopian interpolation weighted mean convergence

A Quasi-Interpolation Satisfying Quadratic Polynomial Reproduction with Radial Basis Functions

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Numerical Mathematics A Journal of Chinese Universities(English Series) 2007年第4期16卷 348-357页

作者： Li Zha renzhong feng Department of Mathematics Beijing University of Aeronautics and Astronautics Beijing 100083 China Key Laboratory of Mathematics Informatics and Behavioral Semantics Ministry of Education

In this paper,a new quasi-interpolation with radial basis functions which satis- fies quadratic polynomial reproduction is constructed on the infinite set of equally spaced data.A new basis function is constructed by making convolution integral with a constructed spline and a given radial basis *** particular,for twicely differ- entiable function the proposed method provides better approximation and also takes care of derivatives approximation.

关键词：二次方程多项式等距处理数据处理

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利于翼型优化设计的超临界翼型参数化方法

北京航空航天大学学报 2011年第3期37卷 368-373页

作者：邓金秋冯仁忠北京航空航天大学数学与系统科学学院北京100191

为减少超临界翼型优化中的设计变量,消除优化结果的不光顺现象、保证C2连续,在优化过程中控制翼型几何特性的变化范围,设计出了由4条首尾相接的有理Bézier曲线表示的超临界翼型的翼型参数化方法,该方法对翼型数据的参数化过程中主要运... 详细信息

为减少超临界翼型优化中的设计变量,消除优化结果的不光顺现象、保证C2连续,在优化过程中控制翼型几何特性的变化范围,设计出了由4条首尾相接的有理Bézier曲线表示的超临界翼型的翼型参数化方法,该方法对翼型数据的参数化过程中主要运用了Bézier曲线拟合算法与SPSA(Simultaneous Perturbation Stochastic Approximation)优化算法,并在Bézier曲线拟合算法中使用了有别于常用方法的数据点参数选择方法.将这种超临界翼型参数化方法与优化算法结合便可实现翼型优化设计,其中的设计变量为21个,优化结果不仅光顺且满足C2条件,通过设定设计变量变化范围便可控制相应的翼型前缘半径、上下弦最高最低点的位置与曲率、尾部契角等几何特征.

关键词：翼型曲线参数化优化

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一种改进的B样条曲线曲面正交距离拟合算法

浙江大学学报（理学版） 2015年第1期42卷 16-20页

作者：余胜蛟冯仁忠北京航空航天大学数学与系统科学学院北京100191 北京航空航天大学数学信息与行为教育部重点实验室北京100191

提出了一种改进的B样条曲线曲面拟合的正交距离算法.在此类算法中,需要求解点投影问题以得到数据点的垂足,考虑到控制顶点对投影的影响,利用泰勒展式对投影算法的初值进行修正,加快了求解点投影问题的速度,从而提高了拟合算法的稳定性... 详细信息

提出了一种改进的B样条曲线曲面拟合的正交距离算法.在此类算法中,需要求解点投影问题以得到数据点的垂足,考虑到控制顶点对投影的影响,利用泰勒展式对投影算法的初值进行修正,加快了求解点投影问题的速度,从而提高了拟合算法的稳定性和效率.数值实验表明,改进算法比修正前的方法更加稳定,与变量投影法及LBFGS算法相比,达到最优解的计算时间更短,迭代步数更少.

关键词： B样条拟合点投影正交距离泰勒展式

一种高精度的修正Hermite-ENO格式

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计算物理 2019年第2期36卷 141-152页

作者：郭子滔冯仁忠北京航空航天大学数学与系统科学学院数学信息与行为教育部重点实验室北京100191

设计一种基于三单元具有六阶精度的修正Hermite-ENO格式(CHENO),求解一维双曲守恒律问题. CHENO格式利用有限体积法进行空间离散,在空间层上,使用ENO格式中的Newton差商法自适应选择模板.在重构半节点处的函数值及其一阶导数值时,利用Ta... 详细信息

设计一种基于三单元具有六阶精度的修正Hermite-ENO格式(CHENO),求解一维双曲守恒律问题. CHENO格式利用有限体积法进行空间离散,在空间层上,使用ENO格式中的Newton差商法自适应选择模板.在重构半节点处的函数值及其一阶导数值时,利用Taylor展开给出修正Hermite插值使其提高到六阶精度,并设计了间断识别法与相应的处理方法以抑制间断处的虚假振荡;在时间层上采用三阶TVD Runge-Kutta法进行函数值及一阶导数值的推进.其主要优点是在达到高阶精度的同时具有紧致性.数值实验表明对一维双曲守恒律问题的求解达到了理论分析结果,是有效可行的.

关键词：双曲守恒律 ENO和WENO格式修正Hermite插值间断识别法