Three-Step Predictor-Corrector of Exponential Fitting Method for Nonlinear Schroedinger Equations

We develop the three-step explicit and implicit schemes of exponential fitting methods. We use the three- step explicit exponential fitting scheme to predict an approximation, then use the three-step implicit exponential fitting scheme to correct this prediction. This combination is called the three-step predictor-corrector of exponential fitting method. The three-step predictor-corrector of exponential fitting method is applied to numerically compute the coupled nonlinear Schroedinger equation and the nonlinear Schroedinger equation with varying coefficients. The numerical results show that the scheme is highly accurate.     

Local polynomial fitting under association

We consider the estimation of multivariate regression functions r(x₁,…,x_d) and their partial derivatives up to a total order p1 using high-order local polynomial fitting. The processes {Y_i,X_i} are assumed to be (jointly) associated. Joint asymptotic normality is established for the estimates of the regression function r and all its partial derivatives up to the total order p. Expressions for the bias and variance/covariance matrix (of the asymptotic distribution) are given.     

Bivariate Free Knot Splines

We consider the least squares approximation of gridded 2D data by tensor product splines with free knots. The smoothing functional to be minimized—a generalization of the univariate Schoenberg functional—is chosen in such a way that the solution of the bivariate problem separates into the solution of a sequence of univariate problems in case of fixed knots. The resulting optimization problem is a constrained separable least squares problem with tensor product structure. Based on some ideas developed by the authors for the univariate case, an efficient method for solving the specially structured 2D problem is proposed, analyzed and tested on hand of some examples from the literature.     

Superconvergence of discontinuous Galerkin finite element method for the stationary Navier‐Stokes equations

This article focuses on discontinuous Galerkin method for the two‐ or three‐dimensional stationary incompressible Navier‐Stokes equations. The velocity field is approximated by discontinuous locally solenoidal finite element, and the pressure is approximated by the standard conforming finite element. Then, superconvergence of nonconforming finite element approximations is applied by using least‐squares surface fitting for the stationary Navier‐Stokes equations. The method ameliorates the two noticeable disadvantages about the given finite element pair. Finally, the superconvergence result is provided under some regular assumptions. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 421–436, 2007     

Linear least squares problems with data over incomplete grids

The paper addresses bivariate surface fitting problems, where data points lie on the vertices of a rectangular grid. Efficient and stable algorithms can be found in the literature to solve such problems. If data values are missing at some grid points, there exists a computational method for finding a least squares spline by fixing appropriate values for the missing data. We extended this technique to arbitrary least squares problems as well as to linear least squares problems with linear equality constraints. Numerical examples are given to show the effectiveness of the technique presented. AMS subject classification (2000)  65D05, 65D07, 65D10, 65F05, 65F20     

几种基于散乱数据拟合的局部插值方法

本文首先针对散乱数据拟合的Shepard方法,结合截断多项式、B样条基函数和指数函数来构造其权函数,使新的权函数具有更高的光滑度和更好的衰减性,并且其光滑性和衰减性可以根据实际需要自由调节,从而提高了曲面的拟合质量．同时还给出一种类似的局部插值方法．另外,本文还基于多重二次插值,结合多元样条的思想,给出了两个局部插值算法．该算法较好地继承了多重二次插值曲面的性质,从而保证了拟合曲面具有好地光顺性和拟合精度．曲面整体也具有较高的光滑性．     

荧光光纤光栅传感特性的实验研究

在载氢掺铒光纤上写入Bragg光纤光栅，得到新型光子学器件-荧光光纤光栅.分别对其Bragg波长（λB）及荧光寿命（τ）的温度（T）及应变（ε）响应特性进行了实验研究，并且给出了λB和τ分别关于（T，ε）的拟合方程.实验结果表明：荧光光纤光栅的λB对T和ε的响应具备一般Bragg光纤光栅的优良特性，测得温度灵敏度为11.1pm /℃，应变灵敏度为1.19pm/με；而且τ对T和ε的响应也具有良好的线性关系，温度灵敏度为0.59 μs/℃，应变灵敏度为6.16 ns/με.实验结论为解决温度应力交叉敏感、实现温度应力的同时监测提供一条新颖的途径.     

子孔径拼接检测非球面的初步研究

使用子孔径拼接技术可以无需补偿器、大口径的辅助镜、全息图等辅助元件实现对大口径、大偏离量、高陡度非球面甚至离轴非球面的检验,而且可以同时获得中高频的相位信息,大大地提高了测量精度,降低了成本。在总结了常用检测非球面方法优缺点的基础上提出了利用圆形子孔径、环形子孔径检测非球面的基本原理,并对其步骤的实现、数学模型的建立和拼接算法的开发进行了分析和研究。结果表明,子孔径拼接检测技术可以作为补偿检验以外的另一种定量测试非球面的手段,可以和其它检测方法相互验证,从而确保检测的准确性和可靠性。     

Numerical analysis for the periodic response of nonsmooth dynamic systems of high-dimension

This paper deals with the numerical solution of high dimensional dynamic systems with nonsmooth characteristics, such as structural elastoplasticity, mechanical clearance and dry friction. As a stable periodic response of the system attracts its transient response, it may be extracted from the transient response somehow. The paper suggests a novel iteration scheme to extrapolate the periodic response from a short time history of its transient response by curve fitting. Compared with the current schemes such as shooting and incremental harmonic balancing, the present scheme makes full use of the information within the transient response and the system characters, then its computation efficiency has increased by an order and the numerical convergence depending on the initial iteration has greatly improved.The subject supported by the Chinese Foundation of Aeronautical Science     

灰色Verhulst模型的样条插值函数的残差修正

本用样条函数对灰色Verhulst模型的残差序列进行插值拟合，然后作用于二阶线性微分方程，并以此修正原模型，得到一种新的预测模型的数值解，提高了拟合的精确度。