C2 Continuous Quartic Hermite Spline Curves with Shape Parameters

In order to relieve the deficiency of the usual cubic Hermite spline curves, the quartic Hermite spline curves with shape parameters is further studied in this work. The interpolation error and estimator of the quartic Hermite spline curves are given. And the characteristics of the quartic Hermite spline curves are discussed. The quartic Hermite spline curves not only have the same interpolation and conti-nuity properties of the usual cubic Hermite spline curves, but also can achieve local or global shape adjustment and C2 continuity by the shape parameters when the interpolation conditions are fixed.     

An algorithm for the computation of the exponential spline

Summary Procedures for the calculation of the exponential spline (spline under tension) are presented in this paper. The procedureexsplcoeff calculates the second derivatives of the exponential spline. Using the second derivatives the exponential spline can be evaluated in a stable and efficient manner by the procedureexspl. The limiting cases of the exponential spline, the cubic spline and the linear spline are included. A proceduregenerator is proposed, which computes appropriate tension parameters. The performance of the algorithm is discussed for several examples.Editor's Note: In this fascile, prepublication of algorithms from the Approximation series of the Handbook for Automatic Computation is continued. Algorithms are published in ALGOL 60 reference language as approved by the IFIP. Contributions in this series should be styled after the most recently published ones     

纵向数据边际模型非参数光滑方法的比较

对于纵向数据边际模型的均值函数, 有很多非参数估计方法, 其中回归样条, 光滑样条, 似乎不相关(SUR)核估计等方法在工作协方差阵正确指定时具有最小的渐近方差. 回归样条的渐近偏差与工作协方差阵无关, 而SUR核估计和光滑样条估计的渐近偏差却依赖于工作协方差阵. 本文主要研究了回归样条, 光滑样条和SUR核估计的效率问题. 通过模拟比较发现回归样条估计的表现比较稳定, 在大多数情况下比光滑样条估计和SUR核估计的效率高.     

A Method for allowing Local Editing of Globally Optimized Splines

The present work describes an algorithm for modifying spline curves in the neighborhood of an editing point while preserving global smoothness properties. Current spline algorithms have either a graphical editing mode in which the user edits local properties of the spline, or a globally optimizing mode, in which the spline coefficients are determined such that overall properties e. g. smoothness, distance to support points, or physical behavior are optimized. With globally optimized splines, editing parameters at one point causes their transmission through the whole spline. Hence, the user has the impression that it is not possible to change the shape of the spline without disturbing the overall behavior. This can be circumvented by forcing the trajectory of the globally optimized curve to lie in the close vicinity of the original curve far away from the edited point. The present work describes an algorithm for local editing of spline curves that are produced by a global optimizer. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Problem of Instability in the Dimensions of Spline Spaces over T-Meshes with T-Cycles

The T-meshes are local modification of rectangular meshes which allow T-junctions. The splines over T-meshes are involved in many fields, such as finite element methods, CAGD etc. The dimension of a spline space is a basic problem for the theories and applications of splines. However, the problem of determining the dimension of a spline space is difficult since it heavily depends on the geometric properties of the partition. In many cases, the dimension is unstable. In this paper, we study the instability in the dimensions of spline spaces over T-meshes by using the smoothing cofactor-conformality method. The modified dimension formulas of spline spaces over T-meshes with T-cycles are also presented. Moreover, some examples are given to illustrate the instability in the dimensions of the spline spaces over some special meshes.     

Hilbertian Approach for Univariate Spline with Tension

In this work, a new approach is proposed for constructing splines with tension. The basic idea is in the use of distributions theory, which allows us to define suitable Hilbert spaces in which the tension spline minimizes some energy functional. Classical orthogonal conditions and characterizations of the spline in terms of a fundamental solution of a differential operator are provided. An explicit representation of the tension spline is given. The tension spline can be computed by solving a linear system. Some numerical examples are given to illustrate this approach.     

Hilbertian Approach for Univariate Spline with Tension

In this work, a new approach is proposed for constructing splines with tension. The basic idea is in the use of distributions theory, which allows us to define suitable Hilbert spaces in which the tension spline minimizes some energy functional. Classical orthogonal conditions and characterizations of the spline in terms of a fundamental solution of a differential operator are provided. An explicit representation of the tension spline is given. The tension spline can be computed by solving a linear system. Some numerical examples are given to illustrate this approach.     

三维Ⅱ型剖分上的样条空间

三维Ⅱ型剖分上的样条空间施锡泉（大连理工大学数学研究所）ＳＰＬＩＮＥＳＰＡＣＥＳＯＮＴＹＰＥ－２ＴＲＩＡＮＧＵＬＡＴＩＯＮＩＮＲ￣３￥ＳｈｉＸｉ－ｑｕａｎ（Ｉｎｓｔ．ｏｆＭａｔｈ．，ＤａｌｉｎＵｎｉｖ．ｏｆＳｃｉ．ａｎｄＴｅｃｈ．）Ａｂｓｔｒａｃｔ：...     

On the Construction of Optimal Monotone Cubic Spline Interpolations

In this paper we derive necessary optimality conditions for an interpolating spline function which minimizes the Holladay approximation of the energy functional and which stays monotone if the given interpolation data are monotone. To this end optimal control theory for state-restricted optimal control problems is applied. The necessary conditions yield a complete characterization of the optimal spline. In the case of two or three interpolation knots, which we call thelocalcase, the optimality conditions are treated analytically. They reduce to polynomial equations which can very easily be solved numerically. These results are used for the construction of a numerical algorithm for the optimal monotone spline in the general (global) case via Newton's method. Here, the local optimal spline serves as a favourable initial estimation for the additional grid points of the optimal spline. Some numerical examples are presented which are constructed by FORTRAN and MATLAB programs.     

Comparative Analysis of Cubic Spline and Kernel Estimation of a Probit Function

The least-squares cubic spline and the kernel estimators produce comparable mean squared errors, although the kernel produces smaller mean squared errors when the variable increases away from 0. Mean squared error increases with an increase in the number of knots (for the cubic spline) or reduced band width (for the kernel estimator). The cubic spline produces smaller mean squared errors when all observations are made at knots than when they are spaced out between knots. Irrespective of the exact form of the probit function g(x), the cubic spline estimator is asymptotically unbiased, while the kernel estimator only converges to g(x) under certain conditions. Moreover, the cubic spline is a smooth function, which is twice differentiable on the interval [0,1].     

混合体积、对数凹序列和B样条函数

本文主要通过样条函数方法研究与之相关的离散几何学和组合学问题.在离散几何学方面主要考虑超立方体切面(cube slicing)体积和混合体(mixed volume)的样条表示,利用B样条函数的几何解释,将超立方体切面问题转化为与之等价的样条函数问题,分别给出Laplace和P′olya关于超立方体切面定理的样条证明,将样条函数与混合体积联系起来,给出一类混合体积的样条解释.利用这种解释可以得到一类具有对数凹性质的组合序列,从而部分地回答了Schmidt和Simion所提出的关于混合体积的公开问题.在组合数学方面主要考虑多种组合多项式与样条函数的关联以及组合序列对数凹性质的样条方法研究.本文借助丰富的样条函数理论,不但验证了离散几何学和组合数学中很多现有的结果,而且得到了一系列离散数学对象的新性质,建立了离散数学问题与具有连续性特质的样条函数之间的内在联系.     

A spline smoothing homotopy method for nonconvex nonlinear programming

Homotopy methods are globally convergent under weak conditions and robust; however, the efficiency of a homotopy method is closely related with the construction of the homotopy map and the path tracing algorithm. Different homotopies may behave very different in performance even though they are all theoretically convergent. In this paper, a spline smoothing homotopy method for nonconvex nonlinear programming is developed using cubic spline to smooth the max function of the constraints of nonlinear programming. Some properties of spline smoothing function are discussed and the global convergence of spline smoothing homotopy under the weak normal cone condition is proven. The spline smoothing technique uses a smooth constraint instead of m constraints and acts also as an active set technique. So the spline smoothing homotopy method is more efficient than previous homotopy methods like combined homotopy interior point method, aggregate constraint homotopy method and other probability one homotopy methods. Numerical tests with the comparisons to some other methods show that the new method is very efficient for nonlinear programming with large number of complicated constraints.     

Splines and Linear Control Theory

In this work, the relationship between splines and the linear control theory has been analyzed. We show that spline functions can be constructed naturally from the control theory. By establishing a framework based on control theory, we provide a simple and systematic way to construct splines. We have constructed the traditional spline functions including polynomial splines and the classical exponential spline. We have also discovered some new spline functions such as the combination of polynomial, exponential and trigonometric splines. The method proposed in this paper is easy to implement. Some numerical experiments are performed to investigate properties of different spline approximations.     

A two-dimensional minimum-derivative spline

We consider the construction of a C ^(1,1) interpolation parabolic spline function of two variables on a uniform rectangular grid, i.e., a function continuous in a given region together with its first partial derivatives which on every partial grid rectangle is a polynomial of second degree in x and second degree in y. The spline function is constructed as a minimum-derivative one-dimensional quadratic spline in one of the variables, and the spline coefficients themselves are minimum-derivative quadratic spline functions of the other variable.     

Bootstrapping for Penalized Spline Regression

We describe and contrast several different bootstrap procedures for penalized spline smoothers. The bootstrap methods considered are variations on existing methods, developed under two different probabilistic frameworks. Under the first framework, penalized spline regression is considered as an estimation technique to find an unknown smooth function. The smooth function is represented in a high-dimensional spline basis, with spline coefficients estimated in a penalized form. Under the second framework, the unknown function is treated as a realization of a set of random spline coefficients, which are then predicted in a linear mixed model. We describe how bootstrap methods can be implemented under both frameworks, and we show theoretically and through simulations and examples that bootstrapping provides valid inference in both cases. We compare the inference obtained under both frameworks, and conclude that the latter generally produces better results than the former. The bootstrap ideas are extended to hypothesis testing, where parametric components in a model are tested against nonparametric alternatives.

Datasets and computer code are available in the online supplements.     

广义非参数回归的B样本贝叶斯估计

本文主要研究广义非参数模型B样条Bayes估计 .将回归函数按照B样条基展开 ,我们不具体选择节点的个数 ,而是节点个数取均匀的无信息先验 ,样条函数系数取正态先验 ,用B样条函数的后验均值估计回归函数 .并给出了回归函数B样条Bayes估计的MCMC的模拟计算方法 .通过对Logistic非参数回归的模拟研究 ,表明B样条Bayes估计得到了很好的估计效果     

How to build all Chebyshevian spline spaces good for geometric design?

In the present work we determine all Chebyshevian spline spaces good for geometric design. By Chebyshevian spline space we mean a space of splines with sections in different Extended Chebyshev spaces and with connection matrices at the knots. We say that such a spline space is good for design when it possesses blossoms. To justify the terminology, let us recall that, in this general framework, existence of blossoms (defined on a restricted set of tuples) makes it possible to develop all the classical geometric design algorithms for splines. Furthermore, existence of blossoms is equivalent to existence of a B-spline bases both in the spline space itself and in all other spline spaces derived from it by insertion of knots. We show that Chebyshevian spline spaces good for design can be described by linear piecewise differential operators associated with systems of piecewise weight functions, with respect to which the connection matrices are identity matrices. Many interesting consequences can be drawn from the latter characterisation: as an example, all Chebsyhevian spline spaces good for design can be built by means of integral recurrence relations.     

Splines in motion — An introduction to MODUS and some unresolved approximation problems

This paper discusses the application of polynomial spline functions in kinematic design problems for planar motion specification. The architecture of a computer-based design system called MODUS, which uses spline functions, is outlined. The testing and evaluation of the system are discussed and future developments are indicated. Mathematical problems and questions in univariate interpolation, which are very relevant to the application of spline functions in this area, are presented for the consideration of the research community in approximation theory.     

Numerical solution of differential-algebraic equations using the spline collocation-variation method

Numerical methods for solving initial value problems for differential-algebraic equations are proposed. The approximate solution is represented as a continuous vector spline whose coefficients are found using the collocation conditions stated for a subgrid with the number of collocation points less than the degree of the spline and the minimality condition for the norm of this spline in the corresponding spaces. Numerical results for some model problems are presented.     

Box－样条曲面的正性条件

研究了Ｂｏｘ－样条曲面的控制点与Ｂｏｘ－样条曲面的正性之间的关系．给出了Ｂｏｘ－样条曲面正性的必要条件、充分条件．由此我们得到了Ｂｏｘ－样条曲面的单调性条件，推广了Ｗ．ＤＡＨＭＥＮ和Ｃ．Ａ．ＭＩＣＣＨＥＬＬＩ在文［２］中给出的Ｂｏｘ－样条曲面的单调性结论