A Generalized Smoother for Linear Ordinary Differential Equations

Ordinary differential equations (ODEs) are equalities involving a function and its derivatives that define the evolution of the function over a prespecified domain. The applications of ODEs range from simulation and prediction to control and diagnosis in diverse fields such as engineering, physics, medicine, and finance. Parameter estimation is often required to calibrate these theoretical models to data. While there are many methods for estimating ODE parameters from partially observed data, they are invariably subject to several problems including high computational cost, complex estimation procedures, biased estimates, and large sampling variance. We propose a method that overcomes these issues and produces estimates of the ODE parameters that have less bias, a smaller sampling variance, and a 10-fold improvement in computational efficiency. The package GenPen containing the Matlab code to perform the methods described in this article is available online.     

Dielectric relaxation of α-cyclodextrin—polyiodide complexes (α-cyclodextrin)2 · LiI3 · I2 · 8H2O and (α-cyclodextrin)2 · Cd0.5 · I5 · 26H2O

The frequency and temperature dependence of real (?′) and imaginary (?″) parts of the dielectric constant of polycrystalline complexes (α-CD)₂ · LiI₃ · I₂ · 8H₂O and (α-CD)₂ · Cd_0.5 · I₅ · 26H₂O (α-CD = α-cyclodextrin) has been investigated over the frequency and temperature ranges of 0–100 kHz and 12–300 K. The dielectric behaviour is described well by Debye type relaxation (α-dispersion). Both systems exhibit an additional Ω dispersion at low frequencies which is attributed to ionic conductance and is much greater in the case of Li due to the greater mobility of cations Li⁺. The temperature dependence of ?′ reveals the existence of two kinds of water molecule in the case of the (α-CD)₂ · Cd_0.5 · I₅ · 26H₂O complex; these can be classified as tiqhtly bound and easily movable water molecules that cause two steps in ?′ versus T plots. In the case of the (α-CD)₂ · LiI₃ · I₂ · 8H₂O complex the water molecules are tightly bound and as a result only one step is observed in these graphs. These finding are also confirmed from the ?″_max versus T plots, which exhibit the same number of steps with ?′, and from calorimetric measurements. The order-disorder transition or the transformation of normal hydrogen bonds to flip-flop type has been observed as a peak in ?″ versus T plots that is more intense and narrow in the case of Li and less high but more broad in Cd. The relaxation time vanes in a α-like curve (from 120 K to 240 K) and rises rapidly for temperatures greater than 240 K, indicating the existence of a new process involving the breaking of hydrogen bonds (normal or flip-flop type). The calculated values of activation energy (0.35–0.62 k_BT_trans) reveal the greater stability of the Li compared with the Cd complex. The starting value of 8.2–8.4 μs for τ is the same as observed in β-CD complexes with guest 4-t-butylbenzyl alcohol (TERB). However, the activation energies of these are greater (1.1–1.7k_BT_trans), indicating greater stability for β-CD complexes.     

Parametric study of the class-shape-refinement-transformation method

The application of a novel parametrization technique to the optimization of aircraft shapes is presented. This class-shape-refinement transformation (CSRT) technique combines an analytical function (class function), a set of Bernstein polynomials (shape function) and a B-spline (refinement function) and can be used to model various aircraft components. It allows for both global and local control of a shape and forms a very efficient and intuitive way of mathematically describing aircraft parts. A parametric study was performed that shows the behaviour of the shape as a function of a number of different parameters, such as total number of shape variables and Bernstein/B-spline coefficient ratio. The CSRT method was used to approximate a typical aircraft wing cross-section and the results showed a very non-linear relationship between the number of shape variables and the error of the approximation, expressed in terms of a correlation factor. This behaviour has been thoroughly analysed. Additionally, optimization results were obtained that show that the CSRT method was successfully coupled to an aerodynamic flow solver. The objective of the optimization runs was to maximize lift-to-drag ratio, but in principle any objective function could be used as long as its input follows from the aerodynamic analysis. The optimization algorithm is capable of largely removing the shock wave on an airfoil at a typical cruise Mach number.     

A shape-preserving approximation by weighted cubic splines

This paper addresses new algorithms for constructing weighted cubic splines that are very effective in interpolation and approximation of sharply changing data. Such spline interpolations are a useful and efficient tool in computer-aided design when control of tension on intervals connecting interpolation points is needed. The error bounds for interpolating weighted splines are obtained. A method for automatic selection of the weights is presented that permits preservation of the monotonicity and convexity of the data. The weighted B-spline basis is also well suited for generation of freeform curves, in the same way as the usual B-splines. By using recurrence relations we derive weighted B-splines and give a three-point local approximation formula that is exact for first-degree polynomials. The resulting curves satisfy the convex hull property, they are piecewise cubics, and the curves can be locally controlled with interval tension in a computationally efficient manner.     

采用B样条有限基集获得等电子系列基态关联能

报导了基于组态相互作用模型对几种双电子原子(离子)系统基态电子关联能的计算方法研究,方法上属于变分法应用的范畴,技巧上采用B样条处理径向基,同时又用Goldman[3]提出的"混合L"法处理角向基,给出He,H^-,Li⁺,Be⁺⁺,B⁺⁺⁺等五种原子(离子)系统的有关计算结果,还与其它理论方法的相应结果作了对比,显示出良好的计算精度和效率     

Parallel algorithms for semi-lagrangian advection

Numerical time step limitations associated with the explicit treatment of advection-dominated problems in computational fluid dynamics are often relaxed by employing Eulerian–Lagrangian methods. These are also known as semi-Lagrangian methods in the atmospheric sciences. Such methods involve backward time integration of a characteristic equation to find the departure point of a fluid particle arriving at a Eulerian grid point. The value of the advected field at the departure point is obtained by interpolation. Both the trajectory integration and repeated interpolation influence accuracy. We compare the accuracy and performance of interpolation schemes based on piecewise cubic polynomials and cubic B-splines in the context of a distributed memory, parallel computing environment. The computational cost and interprocessor communication requirements for both methods are reported. Spline interpolation has better conservation properties but requires the solution of a global linear system, initially appearing to hinder a distributed memory implementation. The proposed parallel algorithm for multidimensional spline interpolation has almost the same communication overhead as local piecewise polynomial interpolation. We also compare various techniques for tracking trajectories given different values for the Courant number. Large Courant numbers require a high-order ODE solver involving multiple interpolations of the velocity field. © 1997 John Wiley & Sons, Ltd.     

Bivariate box splines and smooth pp functions on a three direction mesh

Let S denote the space of bivariate piecewise polynomial functions of degree ? k and smoothness ρ on the regular mesh generated by the three directions (1, 0), (0, 1), (1, 1). We construct a basis for S in terms of box splines and truncated powers. This allows us to determine the polynomials which are locally contained in S and to give upper and lower bounds for the degree of approximation. For $\text{ρ = ?}\text{(2k ? 2)}\text{3}\text{?}$, k ? 2 (3), the case of minimal degree k for given smoothness ρ, we identify the elements of minimal support in S and give a basis for $\text{S}{}_{\mathrm{<mtext>loc</mtext>}}\text{= {f ∈ S:}\text{supp}\text{f ? Ω}}$, with Ω a convex subset of $\text{R}$².     

一类双k次B样条曲面的G1连续性条件

本文针对两个k×k次B样条曲面的节点向量为端点插值、内部是单节点的情形 ,给出它们之间的G1光滑拼接条件 ,同时得到它们的公共边界曲线的控制顶点所要满足的本征方程 .其中本征方程是B样条曲面片所独有的现象 .     

Product integration of logarithmic singular integrands based on cubic splines

This paper is concerned with the practical evaluation of the product integral ∫¹_{? 1}f(x)k(x)dx for the case when k(x) = In|x - λ|, λ? (?1, +1) and f is bounded in [?1, +1]. The approximation is a quadrature rule

where the weights {w_n,n,i} are chosen to be exact when f is given by a linear combination of a chosen set of functions {φ_n,j}. In this paper the functions {φ_n,j} are chosen to be cubic B-splines. An error bound for product quadrature rules based on cubic splines is provided. Examples that test the performance of the product quadrature rules for different choices of the function are given. A comparison is made with product quadrature rules based on first kind Chebyshev polynomials.     

基于正交规范化的小波的有限差分表示

本文研究了在时域内小波的一种表达形式.利用正交规范化,获得了小波的有限差分表示.不仅该形式构造了任意次B样条正交小波.而且在时域中用来直接获得小波滤波器是有效的.