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。     

积分值五次样条拟插值

为了解决连续区间上积分值的函数重构,提出一种直接构造方法.首先利用积分值的线性组合得到结点处函数值的六阶逼近;然后将该近似函数值代入到传统的五次离散样条拟插值算子的泛函值上,得到积分值五次样条插值;最后基于五次样条拟插值的收敛阶,得到该方法对高阶导数逼近的收敛阶.实验结果表明,与传统的积分值样条插值方法相比,该方法简单、有效,可以推广到积分值高次样条拟插值上.     

一类加权有理插值样条曲面及局部约束控制

目的 构造一类新的基于函数值与偏导数值的加权有理插值样条曲面,讨论该样条曲面的相关性质并分析曲面的局部约束控制。方法 一方面,先从x方向构造有理三次插值样条,再从y方向构造二元有理插值样条曲面;另一方面,按相反次序构造另一个二元有理插值样条曲面;最后将两种插值曲面加权得到一类新的有理插值样条曲面。结果 讨论插值曲面的性质,包括基函数、边界性质、积分加权系数的性质以及误差估计。通过选择合适的参数和加权系数,在不改变插值数据的前提下实现对插值区域内的局部约束控制。结论 实验结果表明,新的加权有理插值样条曲面具有良好的约束控制性质。     

基于样条插值函数的离散过程神经网络训练*

为解决过程神经元网络不能直接输入离散样本的问题,提出基于样条插值函数的离散过程神经网络训练算法。首先,将离散过程样本按采样点分段,在采样区间内分别构造样本和权值的分段样条函数;然后,计算样本函数和权函数的乘积在采样区间上的积分,并将此积分值提交给网络的隐层过程神经元;最后,在输出层计算网络输出。分别采用一次、二次、三次样条函数,设计了三种不同的算法。实验结果表明：一次样条计算效率高,逼近能力差;三次样条计算效率低,但逼近能力好;二次样条在计算效率和逼近能力两方面都比较理想。因此,二次样条函数是离散过程神经网络的较好选择。     

基于局部梯度特征的自适应多结点样条图像插值

为了获得质量更好的插值图像,提出了一种新的C^2。连续的支撑区间为（-2,2）的三次多结点样条插值核函数．通过增加结点带来的自由度构造了多结点样条插值公式;分析了在适当的边界条件和约束下三次多结点样条插值的逼近阶;将一维多结点样条插值算法推广到二维,建立了用于图像数据的插值公式;如果忽视图像的局部特征,通常双三次多结点样条插值图像的边缘会有模糊的现象,为此。对多结点样条插值应用逆梯度,得到了自适应多结点样条插值算法;实验所得误差图像和实验所得图像的峰值信噪比也证实了用自适应多结点样条插值算法重建的图像具有更高的质量．     

集多种特性的三次三角伪B样条

目的 为了同时解决传统多项式B样条曲线在形状调控、精确表示常见工程曲线以及构造插值曲线时的不足,提出了一类集多种特性的三次三角伪B样条。方法 首先构造了一组带两个参数的三次三角伪B样条基函数,然后在此基础上定义了相应的参数伪B样条曲线,并讨论了该曲线的特性及光顺性问题,最后研究了相应的代数伪B样条,并给出了最优代数伪B样条的确定方法。结果 参数伪B样条曲线不仅满足C²连续,而且无需求解方程系统即可自动插值于给定的型值点。当型值点保持不变时,插值曲线的形状还可通过自带的两个参数进行调控。在适当条件下,该参数伪B样条曲线可精确表示圆弧、椭圆弧、星形线等常见的工程曲线。相应的代数伪B样条具有参数伪B样条曲线类似的性质,利用最优代数伪B样条可获得满意的插值效果。结论 所提出的伪B样条同时解决了传统多项式B样条曲线在形状调控、精确表示常见工程曲线以及构造插值曲线时的不足,是一种实用的曲线造型方法。     

C1三次插值样条曲线曲面

利用Bézier曲线的端点插值性质,得到了构造三次插值样条曲线曲面的一种改进的基函数——BB基函数。由BB基函数构造了C¹保形三次插值样条曲线;构造了C¹双三次插值样条曲面。     

插值给定数据点的四次PH曲线构造

目的 PH （Pythagorean hodograph）曲线由于具备有理等距曲线、弧长可精确计算等优良的几何性质,广泛应用于数控加工和路径规划等方面。曲线插值是曲线构造的主要手段之一,虽然对PH曲线的Hermite插值方法进行了广泛研究,但插值给定数据点的构造方法仍有待突破,为推广四次PH曲线的应用范围,提出了一种新的四次PH曲线的3点插值问题解决方法。方法 从四次PH曲线的代数充分必要条件出发,在该曲线的Bézier控制多边形中引入辅助控制顶点,指出其中实参数的几何意义,该实参数可作为形状调节因子对构造曲线进行交互。对给定的3个平面型值点进行参数化确定相应的参数值;通过对四次PH曲线一阶导数积分得到曲线的显式表达,其中包含一个待定复常量,将给定的约束点代入曲线的显式表达式得到关于待定复常量的一元二次复方程,求解该复方程并反求Bézier控制顶点得到符合约束条件的四次PH曲线。结果 实验对通过构造插值给定数据点的四次PH曲线进行比较,当形状调节因此改变时,曲线形状可进行有效交互。每次交互得到两条四次PH曲线,通过弧长、弯曲能量、绝对旋转数的计算得到最优曲线,并构造得到PH曲线的等距线。结论 本文方法给定的形状调节参数具有明确的代数意义和几何意义,本文方法易于实现,可有效进行交互。     

二次Bézier样条光顺插值

根据平面曲线的应变能极小原则构造了一条分段二次B啨ｚｉｅｒ样条曲线插值给定的一系列平面型值点列和端点几何约束条件 为了改进插值曲线的整体光顺性 ,提出了确定插值二次B啨ｚｉｅｒ样条曲线在每一个型值点处的最优切矢方向的一种方法     

基于三次样条插值的数值曲率计算

应用基于“not—a—knot”边界条件的三次样条插值函数计算曲率,给出了5点等距条件下求中心节点处数值曲率的计算公式及其收敛性分析,并描述了相应的数值算法。     

Error analysis of the derivative of rational interpolation with a linear denominator

This paper deals with the approximation properties of the derivatives of rational cubic interpolation with a linear denominator. Error expressions of the derivatives of interpolating functions are derived, convergence is established and the optimal error coefficient c _i is proved to be symmetric about the parameters of the rational interpolation. The unified integral form of the error of the second derivative in all subintervals is obtained. A simple expression of the jump of the second derivative at the knots and the conditions for the interpolating function to be C ² in the interpolating interval are given.     

End conditions for interpolatory sextic splines†

In this paper a sextic spline is defined for interpolation at equally spaced knots along with the end conditions required to complete the definition of the spline. These conditions are in terms of given functional values at the knots and lead to uniform convergence of O(h ⁷) throughout the interval of interpolation. The main objective of defining the end conditions for the sextic spline is to use the sextic spline not only for interpolation purposes, but also for the solution of the fifth-order boundary value problem, with the change consistent with the boundary value problem.     

End conditions for interpolatory septic splines

In this paper, septic spline is defined for interpolation at equally spaced knots along with the end conditions required to complete the definition of spline. These conditions are in terms of given functional values at the knots and lead to uniform convergence of O(h ⁸), throughout the interval of interpolation. The main objective of defining the end conditions for septic spline is to use the septic spline for the solutions of sixth-order boundary value problems, being discussed in the subsequent paper.     

Root isolation methods based upon lagrangian interpolation

Two new methods for root isolation are presented. The methods apply to certain real-valued functions of one real variable. Each method takes a function and an interval and returns a set of subintervals in which roots are likely to occur. The methods are proven correct in exact arithmetic, i.e., each root of the function on the interval will be found in one of the subintervals returned. Both methods are based upon Lagrangian interpolation. The first method uses linear interpolation, requires that the function have two continuous derivatives and requires that the user give a bound on the absolute value of the second derivative. The second method uses quadratic interpolation and has similar requirements on the third derivative. The methods use only functional evaluations and the given bounds on derivatives; no other information about the function is required. An implementation is discussed and sample test cases are given which demonstrate the efficiency and utility of the methods. Future research on the use of the methods is also considered.     

The bounded energy optimal control for a class of distributed parameter systems†

The bounded energy optimal control for one-dimensional linear stationary distributed parameter system is solved here. The criterion function is a quadratic functional of the output.

Obtaining the optimal control involves the computation of the solution of a certain non-linear integral equation. The method of solving this integral equation is approximating the kernel of the integral operator by a sequence of degenerate kernels. It is shown that the sequence of approximate solutions of the approximate integral equations converges to the optimal solution; and that the sequence of approximate values of the criterion, converges to the optimal value of the criterion.     

Integral temporal difference learning for continuous-time linear quadratic regulations

In this paper, we propose a temporal difference (TD) learning method, called integral TD learning that efficiently finds solutions to continuous-time (CT) linear quadratic regulation (LQR) problems in an online fashion where system matrix A is unknown. The idea originates from a computational reinforcement learning method known as TD(0), which is the simplest TD method in a finite Markov decision process. For the proposed integral TD method, we mathematically analyze the positive definiteness of the updated value functions, monotone convergence conditions, and stability properties concerning the locations of the closed-loop poles in terms of the learning rate and the discount factor. The proposed method includes the existing value iteration method for CT LQR problems as a special case. Finally, numerical simulations are carried out to verify the effectiveness of the proposed method and further investigate the aforementioned mathematical properties.

     

The approximate solution of initial-value problems for general Volterra integro-differential equations

We study the application of certain spline collocation methods to Volterra integro-differential equations of orderr where ther-th order derivative of the unknown solution occurs also in the kernel of the integral term. The analysis focuses on the question of the optimal discrete convergence order (at the knots of the approximating spline function).     

A Fully Discrete Fast Fourier–Galerkin Method Solving a Boundary Integral Equation for the Biharmonic Equation

We develop a fully discrete fast Fourier–Galerkin method for solving a boundary integral equation for the biharmonic equation by introducing a quadrature scheme for computing the integrals of non-smooth functions that appear in the Fourier–Galerkin method. A key step in developing the fully discrete fast Fourier–Galerkin method is the design of a fast quadrature scheme for computing the Fourier coefficients of the non-smooth kernel function involved in the boundary integral equation. We prove that with the proposed quadrature algorithm, the total number of additions and multiplications used in generating the compressed coefficient matrix for the proposed method is quasi-linear (linear with a logarithmic factor), and the resulting numerical solution of the equation preserves the optimal convergence order. Numerical examples are presented to demonstrate the approximation accuracy and computational efficiency of the proposed method.     

解一类弱间断最优控制问题的一个自适应拟谱方法

针对弱间断最优控制问题,提出一种自适应拟谱方法.利用一些点序列分割时间区间为若干子区间;控制和状态函数使用分段连续多项式逼近;以数值解的收敛性为基础,证明分割时间区间的点序列可以收敛到弱间断点;依据柯西收敛原理,弱间断点位置可以由前述点序列的变化来估计,据此设计误差指示量以调整子区间和逼近多项式次数.在数值算例中,通过与两种拟谱方法比较,所提出方法在精度和效率上都有更好的表现.