Nearly monotone and nearly convex approximation by smooth splines in ,

Given a monotone or convex function on a finite interval we construct splines of arbitrarily high order having maximum smoothness which are “nearly monotone” or “nearly convex” and provide the rate of -approximation which can be estimated in terms of the third or fourth (classical or Ditzian–Totik) moduli of smoothness (for uniformly spaced or Chebyshev knots). It is known that these estimates are impossible in terms of higher moduli and are no longer true for “purely monotone” and “purely convex” spline approximation.     

Polynomial splines and eigenvalue approximations on quantum graphs

A notion of splines is introduced on a quantum graph Γ. It is shown that eigen values of a Hamiltonian on a finite graph Γ can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polynomial splines on Γ. In particular, a bounded set of eigenvalues can be determined using a space of such polynomial splines with a fixed set of singularities. It is also shown that corresponding eigenfunctions can be reconstructed as uniform limits of the same polynomial splines with appropriate fixed set of singularities.     

Perfect spline approximation

Our study of perfect spline approximation reveals: (i) it is closely related to ΣΔ modulation used in one-bit quantization of bandlimited signals. In fact, they share the same recursive formulae, although in different contexts; (ii) the best rate of approximation by perfect splines of order r with equidistant knots of mesh size h is h^r−1. This rate is optimal in the sense that a function can be approximated with a better rate if and only if it is a polynomial of degree <r.The uniqueness of best approximation is studied, too. Along the way, we also give a result on an extremal problem, that is, among all perfect splines with integer knots on , (multiples of) Euler splines have the smallest possible norms.     

用广义交互确认方法选择良好参数进行多元多项式散乱数据自然样条光顺

1．简介多元样条函数在多元逼近中发挥很大作用，已有数量相当多的综合报告和研究论文正式发表，就在1996年6月在法国召开的第三届国际曲线与曲面会议上便有不少多元样条方面的报告，不过总的感觉是仍然缺乏对噪声数据特别是散乱数据的有效光顺方法．李岳生、崔锦泰、关履泰、胡日章等讨论广义调配样条与张量积函数，并用希氏空间样条方法处理多元散乱数据样条插值与光顺，提出多元多项式自然样条，推广了相应一元的结果．我们知道，在样条光顺中有一个如何选择参数的问题，用广义交互确认方法（generalizedcross－validation，以下简称GC…     

样条的机械化求解

本文在逐次分解法的基础上，给出一种样条机械化求解方法．该方法对多项式样条，有理样条乃至更一般样条的研究都是十分有效的．它适用于三角剖分，矩形剖分乃至更一般的代数曲线剖分     

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.     

Sizer for smoothing splines

Smoothing splines are an attractive method for scatterplot smoothing. The SiZer approach to statistical inference is adapted to this smoothing method, named SiZerSS. This allows quick and sure inference as to “which features in the smooth are really there” as opposed to “which are due to sampling artifacts”, when using smoothing splines for data analysis. Applications of SiZerSS to mode, linearity, quadraticity and monotonicity tests are illustrated using a real data example. Some small scale simulations are presented to demonstrate that the SiZerSS and the SiZerLL (the original local linear version of SiZer) often give similar performance in exploring data structure but they can not replace each other completely. Marron’s research was supported by the Dept. of Stat. and Appl. Prob., National Univ. of Singapore, and by the National Science Foundation Grant DMS-9971649. Zhang’s research was supported by the National Univ. of Singapore Academic Research grant R-155-000-023-112. The Editor, the Associate Editor, and the referees are appreciated for their invaluable comments and suggestions that help improve the article significantly.     

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.     

To the theory of optimal splines

On June 18, 2008 at the Plenary Meeting of the International Conference “Differential Equations and Topology” dedicated to the 100th anniversary of Pontryagin, the report [1] was submitted by Isaev and Leitmann. This report in a summary form included a section dedicated to the research of scientists of TsAGI in the field of automation of full life-cycle (i.e. engineering-design-manufacturing, or CAE/CAD/CAM, or CALS-technologies) of wind tunnel models [2]. Within this framework, methods of geometric modeling [3] and [4] were intensively developed, new classes of optimal splines have been built, including the Pontryagin splines and the Chebyshev splines [5], [6], [7] and [8]. This paper reviews some results on the Pontryagin splines. We also give some results on the Lurie splines, that arise in the problem of interpolation of a cylindrical type surface given by the family of table coplanar planes.     

A constructive approach to nodal splines

In the context of local spline interpolation methods, nodal splines have been introduced as possible fundamental functions by de Villiers and Rohwer in 1988. The corresponding local spline interpolation operator possesses the desirable property of reproducing a large class of polynomials. However, it was remarked that their definition is rather intricate so that it seems desirable to reveal the actual origin of these splines. The real source can be found in the Martensenoperator which can be obtained by two-point Hermite spline interpolation problem posed and proved by Martensen [Darstellung und Entwicklung des Restgliedes der Gregoryschen Quadraturformel mit Hilfe von Spline-Funktionen, Numer. Math. 21(1973)70–80]. On the one hand, we will show how to represent the Hermite Martensen spline recursively and, on the other hand, explicitly in terms of the B-spline by using the famous Marsden identity. Having introduced the Martensenoperator, we will show that the nodal spline interpolation operator can be obtained by a special discretization of the occurring derivatives. We will consider symmetric nodal splines of odd degree that can be obtained by our methods in a natural way.     

Shape-Preserving Approximation of Spatial Data

We present a new method for the construction of shape-preserving curves approximating a given set of 3D data, based on the space of quintic like polynomial splines with variable degrees recently introduced in [7]. These splines – which are C ³ and therefore curvature and torsion continuous – possess a very simple geometric structure, which permits to easily handle the shape-constraints.     

Spatial Regression Models,Response Surfaces,and Process Optimization

Abstract

Spatial regression models are developed as a complementary alternative to second-order polynomial response surfaces in the context of process optimization. These models provide estimates of design variable effects and smooth, data-faithful approximations to the unknown response function over the design space. The predicted response surfaces are driven by the covariance structures of the models. Several structures, isotropic and anisotropic, are considered and connections with thin plate splines are reviewed. Estimation of covariance parameters is achieved via maximum likelihood and residual maximum likelihood. A feature of the spatial regression approach is the visually appealing graphical summaries that are produced. These allow rapid and intuitive identification of process windows on the design space for which the response achieves target performance. Relevant design issues are briefly discussed and spatial designs, such as the packing designs available in Gosset, are suggested as a suitable design complement. The spatial regression models also perform well with no global design, for example with data obtained from series of designs on the same space of design variables. The approach is illustrated with an example involving the optimization of components in a DNA amplification assay. A Monte Carlo comparison of the spatial models with both thin plate splines and second-order polynomial response surfaces for a scenario motivated by the example is also given. This shows superior performance of the spatial models to the second-order polynomials with respect to both prediction over the complete design space and for cross-validation prediction error in the region of the optimum. An anisotropic spatial regression model performs best for a high noise case and both this model and the thin plate spline for a low noise case. Spatial regression is recommended for construction of response surfaces in all process optimization applications.     

A Family of Spline Quasi-Interpolants on the Sphere

In this paper, we construct a local quasi-interpolant Q for fitting a function f defined on the sphere S. We first map the surface S onto a rectangular domain and next, by using the tensor product of polynomial splines and 2-periodic trigonometric splines, we give the expression of Qf. The use of trigonometric splines is necessary to enforce some boundary conditions which are useful to ensure the C ² continuity of the associated surface. Finally, we prove that Q realizes an accuracy of optimal order.     

Algorithm for Optimal Triangulations in Scattered Data Representation and Implementation

Scattered data collected at sample points may be used to determine simple functions to best fit the data. An ideal choice for these simple functions is bivariate splines. Triangulation of the sample points creates partitions over which the bivariate splines may be defined. But the optimality of the approximation is dependent on the choice of triangulation. An algorithm, referred to as an Edge Swapping Algorithm, has been developed to transform an arbitrary triangulation of the sample points into an optimal triangulation for representation of the scattered data. A Matlab package has been completed that implements this algorithm for any triangulation on a given set of sample points.     

The local integro cubic splines and their approximation properties

We show the integro cubic splines proposed by Behforooz [1] can be constructed locally by using B-representation of splines. The approximation properties of the local splines are also considered.     

Latent class models for time series analysis

Latent class analysis of time series designed to classify and compare sets of series is discussed. For a particular time series in latent class the data are independently normally distributed with a vector of means, and common variance , that is, . The function of time, , can be represented by a linear combination of low-order splines (piecewise polynomials). The probability density function for the data of a time series is posited to be a finite mixture of spherical multivariate normal densities. The maximum-likelihood function is optimized by means of an EM algorithm. The stability of the estimates is investigated using a bootstrap procedure. Examples of real and artificial data are presented. Copyright © 1999 John Wiley & Sons, Ltd.     

A local Lagrange interpolation method based on cubic splines on Freudenthal partitions

A trivariate Lagrange interpolation method based on cubic splines is described. The splines are defined over a special refinement of the Freudenthal partition of a cube partition. The interpolating splines are uniquely determined by data values, but no derivatives are needed. The interpolation method is local and stable, provides optimal order approximation, and has linear complexity.

     

Spline Techniques for the Numerical Solution of Singular Perturbation Problems

One-dimensional singularly-perturbed two-point boundary-value problems arising in various fields of science and engineering (for instance, fluid mechanics, quantum mechanics, optimal control, chemical reactor theory, aerodynamics, reaction-diffusion processes, geophysics, etc.) are treated. Either these problems exhibits boundary layer(s) at one or both ends of the underlying interval or they possess oscillatory behavior depending on the nature of the coefficient of the first derivative term. Some spline difference schemes are derived for these problems using splines in compression and splines in tension. Second-order uniform convergence is achieved for both kind of schemes. By making use of the continuity of the first-order derivative of the spline function, a tridiagonal system is obtained which can be solved efficiently by well-known algorithms. Numerical examples are given to illustrate the theory.     

On the Characterization of Quadratic Splines

A quadratic spline is a differentiable piecewise quadratic function. Many problems in the numerical analysis and optimization literature can be reformulated as unconstrained minimizations of quadratic splines. However, only special cases of quadratic splines have been studied in the existing literature and algorithms have been developed on a case-by-case basis. There lacks an analytical representation of a general or even convex quadratic spline. The current paper fills this gap by providing an analytical representation of a general quadratic spline. Furthermore, for a convex quadratic spline, it is shown that the representation can be refined in the neighborhood of a nondegenerate point and a set of nondegenerate minimizers. Based on these characterizations, many existing algorithms for specific convex quadratic splines are also finitely convergent for a general convex quadratic spline. Finally, we study the relationship between the convexity of a quadratic spline function and the monotonicity of the corresponding linear complementarity problem. It is shown that, although both conditions lead to easy solvability of the problem, they are different in general.This project was initiated when the first author was visiting the Technical University of Denmark and Erasmus University. The visit was partially funded by the Danish Natural Science Research Council.     

B-splines on 3-D tetrahedron partition in four-directional mesh

It is more difficult to construct 3-D splines than in 2-D case. Some results in the three directional meshes of bivariate case have been extended to 3-D case and corresponding tetrahedron partition has been constructed. The support of related B-splines and their recurrent formulas on integration and differentiation-difference are obtained. The results of this paper can be extended into higher dimension spaces, and can be also used in wavelet analysis, because of the relationship between spline and wavelets.