一种径向基函数插值的ENO格式

将ENO格式和径向基函数插值相结合,提出了求解双曲型偏微分方程的径向基函数插值的ENO方法。该方法依据ENO思想建立自适应模板,在选定的模板上利用径向基函数进行逼近,能够很好地处理具有间断解的问题,消除间断点处数值振荡现象。以一维双曲型偏微分方程为例,对该方法进行了验证,并通过与多项式ENO格式比较,表明该方法更具有优势。     

多层积分值三次样条拟插值

目的 在实际问题中,某些插值问题结点处的函数值往往是未知的,而仅仅知道一些连续等距区间上的积分值。为此提出了一种基于未知函数在连续等距区间上的积分值和多层样条拟插值技术来解决函数重构。该方法称之为多层积分值三次样条拟插值方法。方法 首先,利用积分值的线性组合来逼近结点处的函数值;然后,利用传统的三次B-样条拟插值和相应的误差函数来实现多层三次样条拟插值;最后,给出两层积分值三次样条拟插值算子的多项式再生性和误差估计。结果 选取无穷次可微函数对多层积分值三次样条拟插值方法和已有的积分值三次样条拟插值方法进行对比分析。数值实验印证了本文方法在逼近误差和数值收敛阶均稍占优。结论本文多层三次样条拟插值函数能够在整体上很好的逼近原始函数,一阶和二阶导函数。本文方法较之于已有的积分值三次样条拟插值方法具有更好的逼近误差和数值收敛阶。该方法对连续等距区间上积分值的函数重构具有普适性。     

权重归一化拉格朗日插值及其空间降尺度应用

针对拉格朗日插值法当基函数阶数较高时易出现龙格现象的问题,提出对插值基函数的权重进行归一化,以提高其在高阶插值时的稳定性。拉格朗日插值法简单高效,突出优点是计算迭代过程有规律,宜通过编程来处理大量的数据,但计算结果会在插值区间边缘出现强烈的龙格现象,这种现象在处理较为平稳的气象强迫数据时尤为明显。本文针对这一缺陷对插值基函数进行了归一化,然后利用给定的插值节点得到数值稳定的插值结果。改进方法在驱动陆面过程模型运行的气象要素空间降尺度中进行了验证。几组比较实验证明改进后的方法比原始的拉格朗日插值方法效果更好。     

带权Bernstein基的对偶基函数在等距逼近中的应用

利用带权Bernstein基的对偶基函数,给出了Bernstein基的对偶泛函和平方可积函数的最小二乘逼近算法,并考虑了满足端点高阶约束条件时的情形.将该算法应用于Bézier曲线等距曲线多项式逼近算法中,不仅可以获得显式的同阶Bézier逼近曲线,还可以满足端点高阶约束条件,进一步还可得到有理逼近算法.数值实例以及与...     

Bézier曲线的等距曲线的同次多项式逼近

等距曲线广泛应用工数控机床加工过程、机器人行走路线、刺绣针法生成等工业领域中,与基曲线相比,其表示更为复杂,基本小能用有理曲线来精确表示.为了使等距曲线与CAD/CAM系统更好地相容,基于圆弧的Bézier多项式逼近,提出一种Bézier曲线的等距曲线的同次多项式逼近方法.首先利用Tchebyshev多项式逼近圆弧,并由此得到圆弧的任意次数的Bézier多项式逼近;然后利用上述圆弧逼近的方法去逼近等距曲线的基圆.进而推导出了一种Bézier曲线的等距曲线多项式逼近方法,得到等距逼近曲线是与基曲线次数相同的Bézier曲线.最后通过实例与其他基于圆弧逼近的等距曲线逼近方法进行了比较,结果表明,文中方法与其他方法具有相似的逼近效果,但大大降低了逼近次数.     

基于参数速度逼近的等距曲线有理逼近

该文提出了曲线的参数速度逼近问题 ,指出等距曲线逼近的关键在于参数速度的逼近 ,并用两种方式来实现它 .首先 ,以法矢方向曲线的控制顶点模长为 Bézier纵标构造 Bernstein多项式 ,以它来逼近曲线的参数速度 ,给出了相应的几何方式的等距逼近算法 ,进一步利用法矢方向曲线的升阶获得了高精度逼近 .其次 ,基于参数速度的 L egendre多项式逼近和插值区间端点的 Jacobi多项式逼近 ,导出了保持法矢平移方向的两种代数方式的等距有理逼近算法 .     

双二次Hermite插值与双二次样条插值

在一维插值问题中,如果给定节点处的函数值和一阶导数值,我们来构造分段插值多项式,其整体具有连续的一阶导数,并且使多项式的次数尽可能低.众所周知,一般采用三次分段Hermite插值函数,其逼近阶对于足够光滑的函数为四阶.然而,对于光滑度较差的函数,三次Hermite插值不但达不到最高的逼近阶,而且容易出现多余的拐点.从保     

函数的大范围展开与离散函数的逼近

用插值方法逼近离散函数(以表格给出的函数)是数值分析中最常用的手段。有些多项式插值算子(例如见[1—3])尽管在理论上对u∈Lipα(0<α<1)或u∈C[-1,1]具有多项式最佳逼近阶或“几乎”最佳逼近阶,但是当节点数目很大时,它们即使在计算机上也是不适用的。 本文构造了一种适合于计算机上实施的逼近序列。它的分析结构十分简单,对于逼近节点数目很大的离散函数颇为有效。这种逼近序列误差在索伯列夫范数意义下步步缩     

计算机辅助几何设计中的保形插值理论及应用

本文主要研究计算机辅助几何设计中的分段多项式保形插值理论与算法 ,分段参数多项式保形插值方法及GHI问题 ,参数曲线弧长参数化的混合数值算法与近似方法 ,与给定任意切线多边形相切的保形逼近样条曲线 ,Bézier曲线和 NURBS曲线的等距线生成以及一般参数曲线等距线的保形逼近曲线。本文首先系统地研究了分段多项式的保形插值 ,建立了分段多项式的保形插值理论框架 ,导出了分段三次Hermite插值保形的充要条件 ,构造了一个 C1 连续的分段三次多项式保形插值算法 ,导出了 2 k+1次或 2 k次多项式保凸的充要条件 ,给出了插入内结点的区域…     

基于模算术系数解析的稀疏插值算法

稀疏多元多项式插值是利用多项式的稀疏结构及其给定的插值点信息重构黑盒函数的一种有效策略，被广泛应用于科学和工程领域。传统的基于Prony方法的稀疏插值算法，其复杂度与多项式项数和次数相关，遇到大规模问题时由于执行多个高阶代数运算而效率较低。提出一种新的求解稀疏多元多项式插值问题的算法，核心操作是利用模算术解析单变元多项式的系数，避免了传统方法必需的高阶方程组求解、高次方程求根等。该算法设定一变元为主元，将黑盒多元多项式视为该主元的单变元多项式，通过解析主元的系数多项式在不同插值点处的函数值，进而重构这些系数多项式以恢复整个多元多项式。理论分析和数值实验表明了算法的有效性和可行性。     

多元多项式函数的三层前向神经网络逼近方法

该文首先用构造性方法证明:对任意r阶多元多项式,存在确定权值和确定隐元个数的三层前向神经网络.它能以任意精度逼近该多项式.其中权值由所给多元多项式的系数和激活函数确定,而隐元个数由r与输入变量维数确定.作者给出算法和算例,说明基于文中所构造的神经网络可非常高效地逼近多元多项式函数.具体化到一元多项式的情形,文中结果比曹飞龙等所提出的网络和算法更为简单、高效;所获结果对前向神经网络逼近多元多项式函数类的网络构造以及逼近等具有重要的理论与应用意义,为神经网络逼近任意函数的网络构造的理论与方法提供了一条途径.     

High-Order Convergence of Spectral Deferred Correction Methods on General Quadrature Nodes

It has been demonstrated that spectral deferred correction (SDC) methods can achieve arbitrary high order accuracy and possess good stability properties. There have been some recent interests in using high-order Runge-Kutta methods in the prediction and correction steps in the SDC methods, and higher order rate of convergence is obtained provided that the quadrature nodes are uniform. The assumption of the use of uniform mesh has a serious practical drawback as the well-known Runge phenomenon may prevent the use of reasonably large number of quadrature nodes. In this work, we propose a modified SDC methods with high-order integrators which can yield higher convergence rates on both uniform and non-uniform quadrature nodes. The expected high-order of accuracy is theoretically verified and numerically demonstrated.     

A Simple Regularization of the Polynomial Interpolation for the Resolution of the Runge Phenomenon

A polynomial interpolation based on a uniform grid yields the well-known Runge phenomenon, where maximum error is unbounded for functions with complex roots in the Runge zone. In this paper, we investigate the Runge phenomenon with the finite precision operation. We first show that the maximum error is bounded because of round-off errors inherent to the finite precision operation. Then we propose a simple truncation method based on the truncated singular value decomposition. The method consists of two stages: In the first stage, a new interpolating matrix is constructed using the assumption that the function is analytic. The new interpolating matrix is preconditioned using the statistical filter method. In the second stage, a truncation procedure is applied such that singular values of the new interpolating matrix are truncated if they are equal to or lower than a certain tolerance level. We generalize the method, by analyzing the singular vectors of both the original and new interpolation matrices based on the assumption in the first stage. We show that the structure of the singular vectors makes the first stage essential for an accurate reconstruction of the original function. Numerical examples show that exponential decay of the error can be achieved if an appropriate truncation is chosen.     

Multidimensional Spline Interpolation: Theory and Applications

Computing numerical solutions of household’s optimization, one often faces the problem of interpolating functions. As linear interpolation is not very good in fitting functions, various alternatives like polynomial interpolation, Chebyshev polynomials or splines were introduced. Cubic splines are much more flexible than polynomials, since the former are only twice continuously differentiable on the interpolation interval. In this paper, we present a fast algorithm for cubic spline interpolation, which is based on the precondition of equidistant interpolation nodes. Our approach is faster and easier to implement than the often applied B-Spline approach. Furthermore, we will show how to loosen the precondition of equidistant points with strictly monotone, continuous one-to-one mappings. Finally, we present a straightforward generalization to multidimensional cubic spline interpolation.      

Fehlerabschätzungen für gewisse Näherungskoeffizienten

Some simple error estimations for the approximate coefficients in the expansion of a real function by orthonormal polynomials are given. First we replace the function by an interpolation polynomial of degreen according to anyn+1 interpolation nodes, and in the second case we choose the nodes as the roots of some orthonormal polynomials of degreen+1.     

Newton-Thiele插值方法在图像放大中的应用研究

图像放大一般采用插值方法，而插值基函数的选择直接影响放大图像的效果和实时速度．在分析常见插值方法和图像特点的基础上，提出一种新的图像放大方法，利用Thiele连分式和Newton多项式建立有理插值函数和代数插值函数；并通过实验证明，该方法也是一种有效的图像放大方法．     

A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions

We present a numerical algorithm for the construction of efficient, high-order quadratures in two and higher dimensions. Quadrature rules constructed via this algorithm possess positive weights and interior nodes, resembling the Gaussian quadratures in one dimension. In addition, rules can be generated with varying degrees of symmetry, adaptable to individual domains. We illustrate the performance of our method with numerical examples, and report quadrature rules for polynomials on triangles, squares, and cubes, up to degree 50. These formulae are near optimal in the number of nodes used, and many of them appear to be new.     

关于切触有理插值问题的一种分段方法

将插值节点进行分段,利用分段Hermite插值多项式及相应的多项式,采用线性组合方法得到一般切触有理插值函数的表达式,还可方便地给出无极点的切触有理插值函数的构造方法。通过引入参数方法,给出设定次数类型的切触有理插值问题有解的条件,证明了解的存在唯一性,并给出误差估计公式。实例表明所给方法具有直观、灵活和有效性,便于实际应用。     

Regularization approach to inductive genetic programming

This paper presents an approach to regularization of inductive genetic programming tuned for learning polynomials. The objective is to achieve optimal evolutionary performance when searching high-order multivariate polynomials represented as tree structures. We show how to improve the genetic programming of polynomials by balancing its statistical bias with its variance. Bias reduction is achieved by employing a set of basis polynomials in the tree nodes for better agreement with the examples. Since this often leads to over-fitting, such tendencies are counteracted by decreasing the variance through regularization of the fitness function. We demonstrate that this balance facilitates the search as well as enables discovery of parsimonious, accurate, and predictive polynomials. The experimental results given show that this regularization approach outperforms traditional genetic programming on benchmark data mining and practical time-series prediction tasks