保单调的有理三次样条插值

构造了一种C^1连续的保单调的有理三次插值函数。由于函数表达式中含有调节参数，使得插值曲线更具灵活性。     

Martensen Splines

This paper is concerned with the construction of the fundamental functions associated with a two-point Hermite spline interpolation scheme used by Martensen in the context of the remainder of the Gregory quadrature rule. We derive both a recursive construction and an explicit representation in terms of the underlying B-Splines which can easily be deduced using Marsden’s identity. We can make use of these functions in order to introduce a local interpolation scheme which reproduces all splines. Finally, we examine the error of this interpolant to a sufficiently smooth function and realize that it behaves like in the case of splines of degree n. AMS subject classification (2000) 65D05, 65D07, 41A15     

Adaptive Surface Reconstruction Based on Tensor Product Algebraic Splines

Surface reconstruction from unorganized data points is a challenging problem in Computer Aided Design and Geometric Modeling. In this paper, we extend the mathematical model proposed by Juttler and Felis (Adv. Comput. Math., 17 (2002), pp. 135-152) based on tensor product algebraic spline surfaces from fixed meshes to adaptive meshes. We start with a tensor product algebraic B-spline surface defined on an initial mesh to fit the given data based on an optimization approach. By measuring the fitting errors over each cell of the mesh, we recursively insert new knots in cells over which the errors are larger than some given threshold, and construct a new algebraic spline surface to better fit the given data locally. The algorithm terminates when the error over each cell is less than the threshold. We provide some examples to demonstrate our algorithm and compare it with Jiittler's method. Examples suggest that our method is effective and is able to produce reconstruction surfaces of high quality.AMS subject classifications: 65D17     

Cubic Splines with Minimal Norm

Natural cubic interpolatory splines are known to have a minimal L ₂-norm of its second derivative on the C ² (or W ₂ ² ) class of interpolants. We consider cubic splines which minimize some other norms (or functionals) on the class of interpolatory cubic splines only. The cases of classical cubic splines with defect one (interpolation of function values) and of Hermite C ¹ splines (interpolation of function values and first derivatives) with spline knots different from the points of interpolation are discussed.     

C2 CONTINUOUS NEARLY ARC-LENGTH PARAMETERIZED CURVE

An efficient method for C~2 nearly arc-length parameterized curve is presented. An idea of approximation for the arc-length function of parametric curve which interpolates CAD data points is discussed. The parameterization is implemented by using parameter transformation. Finally, two numerical examples are given..     

Free-knot Splines Approximation of s-monotone Functions

Let I be a finite interval and r,sN. Given a set M, of functions defined on I, denote by ₊ ^s M the subset of all functions yM such that the s-difference ^s y() is nonnegative on I, >0. Further, denote by ₊ ^s W _p ^r , the class of functions x on I with the seminorm x ^(r)L _p 1, such that ^s x0, >0. Let M _n (h _k ):={ _i=1 ⁿ c _i h _k (w _i t– _i )c _i ,w _i , _i R, be a single hidden layer perceptron univariate model with n units in the hidden layer, and activation functions h _k (t)=t ₊ ^k , tR, kN ₀. We give two-sided estimates both of the best unconstrained approximation E( ₊ ^s W _p ^r ,M _n (h _k ))L _q , k=r–1,r, s=0,1,...,r+1, and of the best s-monotonicity preserving approximation E( ₊ ^s W _p ^r , ₊ ^s M _n (h _k ))L _q , k=r–1,r, s=0,1,...,r+1. The most significant results are contained in theorem 2.2.     

Adaptive Free-Knot Splines

This article proposes a function estimation procedure using free-knot splines as well as an associated algorithm for implementation in nonparametric regression. In contrast to conventional splines with knots confined to distinct design points, the splines allow selection of knot numbers and replacement of knots at any location and repeated knots at the same location. This exibility leads to an adaptive spline estimator that adapts any function with inhomogeneous smoothness, including discontinuity, which substantially improves the representation power of splines. Due to uses of a large class of spline functions, knot selection becomes extremely important. The existing knot selection schemes—such as stepwise selection—suffer the difficulty of knot confounding and are unsuitable for our purpose. A new knot selection scheme is proposed using an evolutionary Monte Carlo algorithm and an adaptive model selection criterion. The evolutionary algorithm locates the optimal knots accurately, whereas the adaptive model selection strategy guards against the selection error in searching through a large candidate knot space. The performance of the procedure is examined and illustrated via simulations. The procedure provides a significant improvement in performance over the other competing adaptive methods proposed in the literature. Finally, usefulness of the procedure is illustrated by an application to actual dataset.     

C^3连续的保形插值三角样本曲线

本给出了构造保形插值曲线的三角样条方法，即在每两个型值点之间构造两段三次参数三角样条曲线。所构造的插值曲线是局部的，保形的和C^3连续的而且曲线的形状可由参数调节。     

Approximation of Surfaces by Fairness Bicubic Splines

In this paper we present an approximation method of surfaces by a new type of splines, which we call fairness bicubic splines, from a given Lagrangian data set. An approximating problem of surface is obtained by minimizing a quadratic functional in a parametric space of bicubic splines. The existence and uniqueness of this problem are shown as long as a convergence result of the method is established. We analyze some numerical and graphical examples in order to prove the validity of our method.     

Spatially Adaptive Bayesian Penalized Regression Splines (P-splines)

In this article we study penalized regression splines (P-splines), which are low-order basis splines with a penalty to avoid undersmoothing. Such P-splines are typically not spatially adaptive, and hence can have trouble when functions are varying rapidly. Our approach is to model the penalty parameter inherent in the P-spline method as a heteroscedastic regression function. We develop a full Bayesian hierarchical structure to do this and use Markov chain Monte Carlo techniques for drawing random samples from the posterior for inference. The advantage of using a Bayesian approach to P-splines is that it allows for simultaneous estimation of the smooth functions and the underlying penalty curve in addition to providing uncertainty intervals of the estimated curve. The Bayesian credible intervals obtained for the estimated curve are shown to have pointwise coverage probabilities close to nominal. The method is extended to additive models with simultaneous spline-based penalty functions for the unknown functions. In simulations, the approach achieves very competitive performance with the current best frequentist P-spline method in terms of frequentist mean squared error and coverage probabilities of the credible intervals, and performs better than some of the other Bayesian methods.     

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.     

Classification of Refinable Splines

A refinable spline is a compactly supported refinable function that is piecewise polynomial. Refinable splines, such as the well-known B-splines, play a key role in computer aided geometric design. So far all studies on refinable splines have focused on positive integer dilations and integer translations, and under this setting a rather complete classification was obtained in [12]. However, refinable splines do not have to have integer dilations and integer translations. The classification of refinable splines with noninteger dilations and arbitrary translations is studied in this paper. We classify completely all refinable splines with integer translations and arbitrary dilations. Our study involves techniques from number theory and complex analysis.     

Finite-Part Integrals and Modified Splines

We first state a uniform convergence theorem for finite-part integrals which are derivatives of weighted Cauchy principal value integrals. We then give a two-stage process to modify approximating splines and optimal nodal splines in such a way that the conditions of this theorem are satisfied. Consequently, these modified splines can be used in the numerical evaluation of these finite-part integrals.     

Interpolating Sequences for Besov Spaces

We characterise the interpolating sequences for the Besov spaces B_p and for their multiplier spaces. We also construct linear operators of interpolation.     

凸四边形上一类具有C~2-连续的插值问题的样条解及其应用

本文讨论了一类凸四边形上的插值问题.指出这类插值问题是可解的,其解是分片二元三次多项式,且在凸四边形上是C~2-连续的.我们证明了这类插值问题的解的存在性和唯一性,给出了解样条的分片表达式及其逼近度的估计.最后还给出了一个应用实例和图形显示来说明本方法是可行的.     

Ⅱ型三角剖分上二次双周期样条

本文用B-网方法确定了△_(nm)~((2))剖分上二次双周期样条函数空间的维数,给出了插值条件的几种提法,证明了解的存在唯一性.     

Adaptive Splines and Genetic Algorithms

Most existing algorithms for fitting adaptive splines are based on nonlinear optimization and/or stepwise selection. Stepwise knot selection, although computationally fast, is necessarily suboptimal while determining the best model over the space of adaptive knot splines is a very poorly behaved nonlinear optimization problem. A possible alternative is to use a genetic algorithm to perform knot selection. An adaptive modeling technique referred to as adaptive genetic splines (AGS) is introduced which combines the optimization power of a genetic algorithm with the flexibility of polynomial splines. Preliminary simulation results comparing the performance of AGS to those of existing methods such as HAS, SUREshrink and automatic Bayesian curve fitting are discussed. A real data example involving the application of these methods to a fMRI dataset is presented.