基于多项式拟合算法的EMD端点问题的处理

经验模态分解(EMD)是由Huang等发展的一种新的数据分析方法,但是在利用样条插值获得上下包络过程中存在着棘手的端点问题。文章在解决该问题已有算法的基础上,提出了多项式拟合算法。它利用多项式来拟合临近端点处的极值点序列,再由此多项式求出极值点序列在端点处的近似取值,使得对极值点进行插值的三次样条在端点处不会发生大的摆动。通过对三种算法进行比较,证明了多项式拟合算法可以有效地抑制端点效应。     

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。     

基于对偶变量变分原理和两端动量独立变量的保辛方法

将广义位移和动量同时用拉格朗日多项式近似,并选择积分区间两端动量为独立变量,然后基于对偶变量变分原理导出了哈密顿系统的离散正则变换和对应的数值积分保辛算法．当位移和动量的拉格朗日多项式近似阶数满足一定条件时,可以自然导出保辛算法的不动点格式．通过数值算例讨论了位移和动量采用不同阶次插值所需最少Gauss积分点个数,并讨论了位移插值阶数、动量插值阶数以及Gauss积分点个数对保辛算法精度的影响,说明了上述不动点格式恰好是一种最优格式．     

Bernstein多项式的快速复合算法

在计算机辅助几何设计中,Bernstein多项式的复合是一个重要的研究课题。目前,实现复合的方法主要有Blossoming算法和优化的Blossoming算法。这类方法虽然是数值稳定的,但是计算量很大,存储空间和程序复杂性方面也要求较高,文中基于多项式插值和符号运算,提出了一种新的复合算法。理论分析表明,新算法不但保持了数值稳定性,而且在计算量,存储空间和程序复杂性方面明显优于已有算法。     

有限域上稀疏多元多项式插值算法

稀疏插值是一种降低计算机代数算法时间复杂度的有效方法,在信号处理、压缩感知、结式计算、图像处理等领域都有广泛应用。为了提高稀疏多元多项式插值算法的效率,对Javadi/Monagan稀疏插值算法进行了改进。首先,消除了必须预先给定项数界T的限制,通过计算特定的矩阵行列式,得到插值多项式f的准确项数。然后,消除了必须预先给定次数界D的限制,通过构造辅助函数,利用概率法结合提前终止技术的Cauchy插值法,得到插值多项式f的准确次数,解决了Javadi和Monagan论文中提出的次数界D过高而导致的高计算复杂度的问题。理论分析和实验结果表明了改进算法的优势,特别是在给定的次数界D过高的情况下,相较于Javadi/Monagan算法,改进算法的性能有较大提高。更进一步,由于改进算法无须给定项数界T和次数界D,对于实际问题在利用插值恢复或近似时更具实用性。     

基于插值的Bernstein多项式复合及其曲线曲面应用

在曲线曲面造型中,Bernstein多项式复合被广泛用于许多几何操作,因而具有重要的理论和实际意义.基于多项式插值和符号计算的思想,研究了Bernstein多项式函数复合问题, 并将其应用于曲线曲面的情形.与两种已有方法相比,新方法具有速度快、易于编程实现、占用存储空间少的特点,但数值精度低于基于广义de Casteljau算法的多项式复合结果.     

自适应细分二次多项式响应面算法

复杂模型(或黑箱函数,black-box function)的全局近似方法广泛应用于模型简化、灵敏度分析和参数(或控制)优化中。现有的单一响应面存在着采样估值与近似精度之间的矛盾,因此提出自适应细分二次多项式响应面算法:在构造初始响应面基础上,对设计域进行递归分割,每步选择最粗糙单元,沿着最粗糙维度方向进行自适应细分。通过实验测试,该方法以多响应面来近似黑箱函数,从而达到采样点少而响应面精度高的目标。     

利用多项式拟合测量飞机油箱油量

飞机飞行中,测量剩余油量是关乎飞行安全的重要工作。以往采用的查表法存在存储数据量过大的问题,而插值法的计算又过于复杂,为此,提出了一种基于多项式数据拟合的油箱剩余油量测量方法,该方法通过存储多项式系数减少数据存储量,通过将多元插值计算改为多项式计算降低运算量。实验中,首先通过地面实验数据拟合获取多项式参数,再利用飞机飞行中获取的测量数据进行计算,所得结果与油箱内剩余油量的实际测量结果相比较,测量误差满足使用要求。通过理论和实验分析,在多项式拟合的阶数选择上明确了使用三阶多项式的合理性。该方法有较好的实用性,已在实践中应用。     

基于局部多项式的自适应图像放大

为了弥补传统图像放大算法的锯齿效应,统计分析了图像像素变化特点,将图像分成平滑、纹理和边缘,对平滑和纹理区根据其局部区域的同态性,提出了基于局部多项式逼近的图像放大：运用置信区间交集法则将图像分割为若干同态区域;对区域分别进行多项式拟合,并对拟合函数进行过采样实现区域放大;运用局部平滑消除边缘锯齿效应。实验结果表明,该算法充分利用了图像的结构信息,相比于传统的插值放大算法和已有局部自适应插值算法提高了图像的视觉效果与适用性。     

多项式插值法在工程计算中应用

本文叙述了多项式插值方法在工程计算中应用。经过论证说明该方法与常用的牛顿插值法和拉格朗日插值法构造的插值多项式是相同的。该方法可作为经验曲线的近似公式，在密集性插值计算中。具有速度快、可反向求值、计算方法简单、引入的舍尾误差小等优点。     

Error analysis of the ultraspherical spectral-collocation method for high-index IAEs

The subject of integral-algebraic equations (IAEs) with index?>?2 is new in the literature and there are a few available results which investigate these systems. We will deal here with a particular class of IAEs of index-3 and analyse ultraspherical polynomials collocation method as well as convergence analysis for the numerical solution of these systems. Our error analysis of collocation method is based on polynomial approximation theory and results related to the interpolation error in Sobolev space. Numerical examples illustrate the theoretical results.     

Nonlinear model order reduction with low rank tensor approximation

In this article, two methods of model order reduction based on the low rank approximation of tensor are introduced for the large scale nonlinear problem. We first introduce some definitions and results on tensor extended from matrix theory. Then we show how the general nonlinear system can be converted into the low rank form we treated in this research. We put the model order reduction of it in two frameworks, that is, polynomial framework and moment‐matching framework. In these two frameworks we construct the algorithms correspondingly, and analyze properties of these algorithms, including the preservation of stability, and moment‐matching properties. Next the priorities of these algorithms are presented. Finally we setup several numerical experiments to validate the effectiveness of the algorithms.     

Rational Pythagorean-hodograph space curves

A method for constructing rational Pythagorean-hodograph (PH) curves in R³ is proposed, based on prescribing a field of rational unit tangent vectors. This tangent field, together with its first derivative, defines the orientation of the curve osculating planes. Augmenting this orientation information with a rational support function, that specifies the distance of each osculating plane from the origin, then completely defines a one-parameter family of osculating planes, whose envelope is a developable ruled surface. The rational PH space curve is identified as the edge of regression (or cuspidal edge) of this developable surface. Such curves have rational parametric speed, and also rational adapted frames that satisfy the same conditions as polynomial PH curves in order to be rotation-minimizing with respect to the tangent. The key properties of such rational PH space curves are derived and illustrated by examples, and simple algorithms for their practical construction by geometric Hermite interpolation are also proposed.     

采用牛顿插值的多项式因式分解算法的设计与实现

基于Kronecker所提供的一元多项式因式分解的构造算法、一元整系数多项式在整数环上因式分解理论,利用牛顿向前差分插值算法代替拉格朗日插值算法,把有理域上一元高次多项式因式分解化为在整数环上的因式分解,得到了整数环上的一元多项式因式分解的构造性算法,给出了具体实现过程。     

Efficient methods for highly oscillatory integrals with weak and Cauchy singularities

In this paper, we present two fast and accurate numerical schemes for the approximation of highly oscillatory integrals with weak and Cauchy singularities. For analytical kernel functions, by using the Cauchy theorem in complex analysis, we transform the integral into two line integrals in complex plane, which can be calculated by some proper Gauss quadrature rules. For general kernel functions, the non-oscillatory and nonsingular part of the integrand is replaced by a polynomial interpolation in Chebyshev points, and the integral is then evaluated by using recurrence relations. Furthermore, several numerical experiments are shown to verify the validity of such methods.     

基于二元Lagrange插值多项式的门限方案

针对基于一元Lagrange插值多项式的门限方案中存在的安全性不足及应用领域受限问题,通过研究现有的门限方案和实数域上的二元Lagrange插值理论,在有限域的基础上,提出一种基于二元Lagrange插值多项式的门限方案。给出了方案的构造及其数值算例,证明了方案的合理性和可行性。将该方案与基于一元Lagrange插值多项式的门限方案进行对比分析,表明新的方案中子秘密丢失所造成的损失更低、合谋难度更大,方案的安全性更高。同时,该方案可以拓宽门限方案的应用领域。     

B-spline surface fitting by iterative geometric interpolation/approximation algorithms

Recently, the use of $B$-spline curves/surfaces to fit point clouds by iteratively repositioning the $B$-spline’s control points on the basis of geometrical rules has gained in popularity because of its simplicity, scalability, and generality. We distinguish between two types of fitting, interpolation and approximation. Interpolation generates a $B$-spline surface that passes through the data points, whereas approximation generates a $B$-spline surface that passes near the data points, minimizing the deviation of the surface from the data points. For surface interpolation, the data points are assumed to be in grids, whereas for surface approximation the data points are assumed to be randomly distributed. In this paper, an iterative geometric interpolation method, as well as an approximation method, which is based on the framework of the iterative geometric interpolation algorithm, is discussed. These two iterative methods are compared with standard fitting methods using some complex examples, and the advantages and shortcomings of our algorithms are discussed. Furthermore, we introduce two methods to accelerate the iterative geometric interpolation algorithm, as well as a method to impose geometric constraints, such as reflectional symmetry, on the iterative geometric interpolation process, and a novel fairing method for non-uniform complex data points. Complex examples are provided to demonstrate the effectiveness of the proposed algorithms.