单纯形上校正高斯-勒让德求积公式

基于高斯-勒让德求积公式余项,给出相应的校正积分公式,提高了至少两阶代数精度。通过坐标变换将三角形、四面体区域变成正方形、立方体积分区域,把校正高斯求积公式推广到高维单纯形上多重积分的计算。通过与二维三角形单元和三维四面体单元上的Hammer求积公式比较发现,校正求积公式的精度非常高,能更快收敛到积分真值,在工程实际中具有较大的应用价值。     

多维超奇异积分的近似计算

多维超奇异积分在弹性力学和电磁场的散射问题等诸多工程领域中有广泛应用.考虑构造二维、三维面型超奇异积分的求积公式,同时提高误差精度.利用复合矩形求积公式在划分的N个子区间内近似计算被积函数中无奇异性的部分,剩余部分通过超奇异积分的解析式求解.根据外推思想,构造一维超奇异积分的修正复合矩形求积公式.最后将带有外推的求积公式推广到二维、三维面型超奇异积分中.文章结尾的数值算例验证了方法的可行性.     

一类带导数的数值求积公式的渐近性质

文章给出了一类带导数的数值求积公式中,当积分区间长度趋向于零时,微分平均值ξ的渐近性质.提出了对应于该式的校正公式,它有较高的代数精度.     

Euler-Maclaurin数值求积公式的渐近性质

文章给出了Euler-Maclaurin数值求积公式中,当积分区间长度趋向于零时,微分平均值ξ的渐近性质．提出了对应于该式的校正公式,它有较高的代数精度．     

非线性延迟积分微分方程线性多步法的渐近稳定性

本文研究了非线性延迟积分微分方程线性多步法的渐近稳定性.证明了在约束网格下,带有复合求积公式A-稳定的线性多步法能够保持解析解的渐近稳定性.文章最后,数值试验验证了本文的结论.     

高振荡非齐次动力系统的有效数值算法

为实现高振荡问题模型方程的有效数值求解,基于高振荡积分的渐进积分算法,针对随时间高频率振荡的非齐次线性动力系统给出有效的数值算法.基于变分常数公式将非齐次动力系统重新表示为指数形式,利用Magnus积分方法求解指数部分,利用渐进积分算法求解高振荡的积分项.数值实验表明：该算法求解精度随振荡频率的增大而提高,且简单易用,也可以容易推广到多个方程的情形.     

Fredholm积分方程的正则化GMRES算法

利用数值求积公式,对二维第1类Fredholm积分方程进行离散处理,引入正则化GMRES算法,将离散后的积分方程转化为离散适定问题,通过广义极小残余算法得到其数值解。数值模拟结果表明,正则化GMRES算法求解二维第1类Fredholm积分方程计算速度快、精度高。     

广义高斯求积公式的渐进计算与数表

近年来,人们将正交多项式的理论推广到了σ-正交多项式．这一推广导出了具有高阶代数精度的广义高斯求积公式．本文中,我们提出一种计算σ-正交多项式零点的高效迭代方法以及计算广义高斯求积公式的科茨系数的简单方法．对于几种常用的权函数,我们还给出求积公式的若干高精度数值结果．     

用最优系数法计算多重积分的表

本表的目的是为了用H.M.柯诺波夫所提出的最优系数法计算多重积分而准备的。用这个方法计算多重积分的公式为■其中a_1,…,a_s是整数,N是求积公式的节点     

基于Lagrange插值多项式拟合的力学系统的变分积分子

变分积分子是通过直接离散变分原理得到的一类特殊的动力学系统的数值差分格式,较之传统差分格式呈现出明显的计算优越性.由离散Euler-Lagrange方程的形式可知,变分积分子的构造过程最终归结为计算离散Lagrange函数的偏导数,其中离散Lagrange函数是Lagrange函数在单个时间步长的积分,通常由经典求积公式近似得到.根据离散Lagrange函数的积分表达式,解析计算其偏导数会随之衍生一个新的且与连续Euler-Lagrange方程密切关联的积分,因此,构造变分积分子就可以不再以通过经典求积公式得到的具体形式的离散Lagrange函数为前提,而是可以直接基于一组离散结点近似新衍生的积分.在这些离散结点处,如果进一步让系统的拟合轨迹严格满足Euler-Lagrange方程,即运动方程,那么新的积分自动为零,相应地,计算离散Lagrange函数的偏导数就简化为计算连续Lagrange函数关于速度变量的偏导数.这种新的构造方式同时结合了连续和离散的Euler-Lagrange方程,不仅让最终得到的差分格式仍然继承了变分积分子特有的优越计算性能,而且在同阶精度的情况下具有更小的局部误差.     

Comparison of some numerical quadrature formulas for weakly singular periodic Fredholm integral equations

Several numerical quadrature formulas that are used in the quadrature method for the numerical solution of periodic Fredholm integral equations are analyzed and precise asymptotic expansions for their errors are derived. All of these formulas are based on the trapezoidal rule with equidistant abscissas. They are compared with respect to their computational cost, accuracy, and efficiency when used in conjunction with the Richardson extrapolation. On the basis of this comparison it is concluded that the formula developed in [4] is the most advantageous. A numerical example is appended.     

Inversion of Linear Space-Time Transformations in Bounded Domains

General formulas for pseudoinversion of linear integral transformations in bounded space-time domains are constructed. The input and output of transformations are given in continuous-discrete forms. The conditions of accuracy and uniqueness of the transformations are studied.     

基于模糊自适应控制的全向移动机器人的运动控制研究

由于四轮驱动全向移动机器人轮系分布的特点,四轮之间存在耦合关系,在运行过程中,机器人整体运动的稳定性及控制精度都不佳。针对此问题,本文设计一种基于模糊自适应控制器的误差修正方法,结合模糊控制和PD控制,在线对机器人体进行误差修正,并将整体误差按轮系结构分布合理分配到单个轮子上,从而将整体的误差修正转化为单个轮子的误差修正。通过在Matlab-Simulink环境下仿真实验表明,在使用模糊自适应控制器进行误差修正后,机器人对线速度及角速度的跟随性明显提高,改善了机器人运动控制的精度。     

Efficient and Accurate Computation of Electric Field Dyadic Green’s Function in Layered Media

Concise and explicit formulas for dyadic Green’s functions, representing the electric and magnetic fields due to a dipole source placed in layered media, are derived in this paper. First, the electric and magnetic fields in the spectral domain for the half space are expressed using Fresnel reflection and transmission coefficients. Each component of electric field in the spectral domain constitutes the spectral Green’s function in layered media. The Green’s function in the spatial domain is then recovered involving Sommerfeld integrals for each component in the spectral domain. By using Bessel identities, the number of Sommerfeld integrals are reduced, resulting in much simpler and more efficient formulas for numerical implementation compared with previous results. This approach is extended to the three-layer Green’s function. In addition, the singular part of the Green’s function is naturally separated out so that integral equation methods developed for free space Green’s functions can be used with minimal modification. Numerical results are included to show efficiency and accuracy of the derived formulas.     

Color correction for tone mapping

Tone mapping algorithms offer sophisticated methods for mapping a real-world luminance range to the luminance range of the output medium but they often cause changes in color appearance. In this work we conduct a series of subjective appearance matching experiments to measure the change in image colorfulness after contrast compression and enhancement. The results indicate that the relation between contrast compression and the color saturation correction that matches color appearance is non-linear and smaller color correction is required for small change of contrast. We demonstrate that the relation cannot be fully explained by color appearance models. We propose color correction formulas that can be used with existing tone mapping algorithms. We extend existing global and local tone mapping operators and show that the proposed color correction formulas can preserve original image colors after tone scale manipulation.     

FFT-based methods for nonlinear image restoration in confocal microscopy

Recently we developed a new method for attenuation correction in 3D imaging by a confocal scanning laser microscope (CSLM) in the (epi)fluorescence mode. The fundamental element in our approach consisted of multiplying the measured fluorescent intensity by a correction factor involving a convolution integral of this intensity, which can be computed efficiently by the fast Fourier transform (FFT). The resulting algorithm is one or two orders of magnitude faster than an existing iterative method, but it was found to have a somewhat smaller accuracy. In this paper we improve on this latter point by reformulating the problem as a statistical estimation problem. In particular, we derive first-order-moment and cumulant estimators leading to a nonlinear integral equation for the unknown fluorescent density, which is solved by an iterative method in which in each step a discrete convolution is performed by using the FFT. We find that only a few iterations are needed. It is shown that the estimators proposed here are more accurate than the existing iterative method, while they retain the advantage in computational efficiency of the FFT-based approach.     

Higher-order correction to the FDTD method based on the integral form of Maxwell’s equations

We propose a higher-order correction to the finite-difference time-domain (FDTD) method based on the integral form of Maxwell’s equations. We calculate the errors between the numerical and analytic solutions. Numerical solutions are obtained by the original method and our corrected FDTD method to show that the accuracy and reliability of our corrected FDTD method is superior to that of the original FDTD method.     

基于眨眼修正卡尔曼滤波的人眼跟踪研究

眼睛运动容易受到头部姿势变化、外界仿真干扰、实际光照条件等影响,已有眼部跟踪算法的准确率、鲁棒性较低。为此,提出一种基于眨眼修正卡尔曼滤波的人眼跟踪算法。采用垂直积分投影函数和水平积分投影函数得到人脸图像的眼睛区域,运用眼睛区域的颜色熵消除不相关因素,定位出瞳孔的位置,用卡尔曼滤波进行实时眼部跟踪,结合眨眼检测实时修正跟踪结果。实验结果表明,该算法准确率较高,实时性较好。     

基于积分图像的快速模板匹配

人脸检测中，模板匹配前往往需要对图像窗口进行灰度分布标准化，而灰度分布标准化则要先算出图像窗口的灰度均值及方差。在积分图像基础上，提出和应用平方积分图像，实现了一种计算速度与图像窗口大小无关的快速灰度均值和方差算法。应用这种快速算法，结合对灰度分布标准化近似计算公式、相关系数和平均偏差计算公式的变换，以厦灰度分布标准化处理时省略拷贝图像窗口及窗口灰度值变换等不必要的步骤，大大提高了模板匹配速度。人脸检测实验证明这种快速模板匹配算法是有效的。