Spline methods in geodetic approximation problems

This paper is concerned with spline methods in a reproducing kernel Hilbert space consisting of functions defined and harmonic in the outer space of a regular surface (e.g. sphere, ellipsoid, telluroid, geoid, (regularized) earth's surface). Spline methods are used to solve interpolation and smoothing problems with respect to a (fundamental) system of linear functional giving information about earth's gravity field. Best approximations to linear functionals are discussed. The spline of interpolation is characterized as the spline of best approximation in the sense of an appropriate (energy) norm.     

A stochastic framework for recursive computation of spline functions: Part II,smoothing splines

Many of the optimal curve-fitting problems arising in approximation theory have the same structure as certain estimation problems involving random processes. We develop this structural correspondence for the problem of smoothing inaccurate data with splines and show that the smoothing spline is a sample function of a certain linear least-squares estimate. Estimation techniques are then used to derive a recursive algorithm for spline smoothing.     

散乱数据的多项式自然样条光顺与广义插值

由于理论与实践的重要性,在多元插值方面有相当多的工作,如[1]-[11]。目前以箱样条(box splines),光滑余因子与B网方法以及薄板样条与径函数(radial basis function)方法比较活跃。前者具有良好的性质和丰富的结构,很快成为一个活跃的研究方向,最近更在小波(wavelet)变换理论研究上发挥了作用。但是,它一般只处理规则分划的问题,不能做多元散乱数据的插值。     

多元散乱数据二步拟合法及其误差估计

多元数据曲面拟合的早期结果,主要在研究格子点的插值问题上,其方法是张量积插值或利用再生核希氏空间理论给出解的构造。[1]系统地总结了1976年以前的研究概况,[2]则为全平面上的薄板样条是一元样条到多元样条非张量积形式的推广。它是基于再生核的明显表示,但对一般的泛函来说,要得到再生核通常是很困难的。最近,[4]避开这一实质性困难,利用Lagrange恒等式,Euler方程及最优插值的特征定理给出了一     

单调光滑函数的保凸插值方法

习知,在二相渗流力学中,毛细管压力曲线 P_C(S_W)=P_(NW)-P_W是很重要的,式中S_W表示润湿相饱和浓度.这个函数没有简单的解析表达式,但据实验分析,它是单调光滑函数,通常有一个拐点,其离散型值由实验确定.根据Ritz原理,用有限元方法解二相渗流问题,对毛细管压力曲线,可采用单调光滑且保持型值的凸凹性的插值函数.类似的插值问题在数值分析中是常见的,本文就是研究这类插值问题.为确定起见,我们讨论递增函数.这些方法不难移到递减的情形.以上插值问题的一般提法是:     

样条型矩阵有理插值

The matrix valued rational interpolation is very useful in the partial realization problem and model reduction for all the linear system theory. Lagrange basic functions have been used in matrix valued rational interpolation. In this paper, according to the property of cardinal spline interpolation, we constructed a kind of spline type matrix valued rational interpolation, which based on cardinal spline. This spline type interpolation can avoid instability of high order polynomial interpolation and we obtained a useful formula.     

Refined-mesh interpolation method for functions with a boundary-layer component

Linear and quadratic spline interpolation methods for a one-variable function with a boundary-layer component are examined. It is shown that the interpolation method for such a function leads to considerable errors when applied on a uniform mesh. The error of linear and quadratic spline interpolations on meshes that are refined in the boundary layer is estimated. Numerical results are presented.     

W_2~m空间中样条插值算子与线性泛函的最佳逼近

In this paper,the convergency of spline interpolation operators is obtained,these spline operators are determined by linear differential operators and constraint functionals.The errors of the interpolating spline with EHB fanctionals are estimated.The best approximation of linear functionals on W2^n spaces are investigated,which let to a useful computational method for the approximation solution of higher order linear differential equations with multipoint boundary value conditions.     

散乱数据带自然边界条件二元样条光顺及数值微分

考虑一种新的散乱数据带自然边界二元样条光顺问题.根据样条变分理论和Hilbert空间样条函数方法,构造出了显式的二元带自然边界光顺样条解,其表达式简单且系数可以由系数矩阵对称正定的线性方程组确定.证明了解的存在和唯一性,讨论了收敛性和误差估计.并由此得到一种新的基于散乱数据上的正则化二元数值微分的方法.最后,给出了一些数值例子对方法进行了验证.     

Wavelet methods for continuous-time prediction using Hilbert-valued autoregressive processes

We consider the prediction problem of a continuous-time stochastic process on an entire time-interval in terms of its recent past. The approach we adopt is based on the notion of autoregressive Hilbert processes that represent a generalization of the classical autoregressive processes to random variables with values in a Hilbert space. A careful analysis reveals, in particular, that this approach is related to the theory of function estimation in linear ill-posed inverse problems. In the deterministic literature, such problems are usually solved by suitable regularization techniques. We describe some recent approaches from the deterministic literature that can be adapted to obtain fast and feasible predictions. For large sample sizes, however, these approaches are not computationally efficient.With this in mind, we propose three linear wavelet methods to efficiently address the aforementioned prediction problem. We present regularization techniques for the sample paths of the stochastic process and obtain consistency results of the resulting prediction estimators. We illustrate the performance of the proposed methods in finite sample situations by means of a real-life data example which concerns with the prediction of the entire annual cycle of climatological El Niño-Southern Oscillation time series 1 year ahead. We also compare the resulting predictions with those obtained by other methods available in the literature, in particular with a smoothing spline interpolation method and with a SARIMA model.     

A simple smoothing spline,III

Summary  Computational methods for spline smoothing are studied in the context of the linear smoothing spline. Comparisons are made between two efficient methods for computing the estimator using band-limited basis functions and the Kalman filter. In particular, the Kalman filter approach is shown to be an efficient method for computing under the Kimeldorf-Wahba representation for the estimator. Run time comparisons are made between band-limited B-spline and Kalman filter based algorithms.     

Mixed interpolating-smoothing splines and the ν-spline

In their monograph, Bezhaev and Vasilenko have characterized the “mixed interpolating-smoothing spline” in the abstract setting of a Hilbert space. In this paper, we derive a similar characterization under slightly more general conditions. This is specialized to the finite-dimensional case, and applied to a few well-known problems, including the ν-spline (a piecewise polynomial spline in tension) and near-interpolation, as well as interpolation and smoothing. In particular, one of the main objectives in this paper is to show that the ν-spline is actually a mixed spline, an observation that we believe was not known prior to this work. We also show that the ν-spline is a limiting case of smoothing splines as certain weights increase to infinity, and a limiting case of near-interpolants as certain tolerances decrease to zero. We conclude with an iteration used to construct curvature-bounded ν-spline curves.     

An energy-minimization framework for monotonic cubic spline interpolation

This paper describes the use of cubic splines for interpolating monotonic data sets. Interpolating cubic splines are popular for fitting data because they use low-order polynomials and have C² continuity, a property that permits them to satisfy a desirable smoothness constraint. Unfortunately, that same constraint often violates another desirable property: monotonicity. It is possible for a set of monotonically increasing (or decreasing) data points to yield a curve that is not monotonic, i.e., the spline may oscillate. In such cases, it is necessary to sacrifice some smoothness in order to preserve monotonicity.The goal of this work is to determine the smoothest possible curve that passes through its control points while simultaneously satisfying the monotonicity constraint. We first describe a set of conditions that form the basis of the monotonic cubic spline interpolation algorithm presented in this paper. The conditions are simplified and consolidated to yield a fast method for determining monotonicity. This result is applied within an energy minimization framework to yield linear and nonlinear optimization-based methods. We consider various energy measures for the optimization objective functions. Comparisons among the different techniques are given, and superior monotonic C² cubic spline interpolation results are presented. Extensions to shape preserving splines and data smoothing are described.     

Constrained interpolation and smoothing

Numerical and theoretical questions related to constrained interpolation and smoothing are treated. The prototype problem is that of finding the smoothest convex interpolant to given univariate data. Recent results have shown that this convex programming problem with infinite constraints can be recast as a finite parametric nonlinear system whose solution is closely related to the second derivative of the desired interpolating function. This paper focuses on the analysis of numerical techniques for solving the nonlinear system and on the theoretical issues that arise when certain extensions of the problem are considered. In particular, we show that two standard iteration techniques, the Jacobi and Gauss-Seidel methods, are globally convergent when applied to this problem. In addition we use the problem structure to develop an efficient implementation of Newton's method and observe consistent quadratic convergence. We also develop a theory for the existence, uniqueness, and representation of solutions to the convex interpolation problem with nonzero lower bounds on the second derivative (strict convexity). Finally, a smoothing spline analogue to the convex interpolation problem is studied with reference to the computation of convex approximations to noisy data.     

一种广义插值法

本文考虑一种广义插值问题，插值条件为小区间上的积分值，以弥补现有的插值方法在L2空间不再适用的不足，除了多项式插值外，还讨论了两种一次样条插值方法。     

关于基样条插值公式

1.引言我们知道,只要构造出基底函数L(x),便得基数型插值公式:     

Multivariate Differences, Polynomials, and Splines

We generalize the univariate divided difference to a multivariate setting by considering linear combinations of point evaluations that annihilate the null space of certain differential operators. The relationship between such a linear functional and polynomial interpolation resembles that between the divided difference and Lagrange interpolation. Applying the functional to the shifted multivariate truncated power produces a compactly supported spline by which the functional can be represented as an integral. Examples include, but are not limited to, the tensor product B-spline and the box spline.     

基于数学方法的不完全投影数据估计

主要对不完全投影图像恢复问题中不完全投影数据进行了研究.对于投影数据不完全时,运用线性插值法和样条插值法在角度缺失的地方进行数据估计.最后通过仿真实验,将得到的估计投影图像与原始投影图像进行比较并分析.实验结果表明:在恢复投影图像的过程当中,采用线性插值误差更小,恢复的投影图像的精度更高,并且恢复速度快.     

分维权重样条插值预测算法及应用

基于高维数据预测方法的应用,提出一种分维权重样条插值预测算法.通过高维数据的各维,建立样本各维数据与对应权重的网络结构关系,网络的结点个数与样本的个数无关.通过训练样本各维权重所满足的线性方程组得到各维的权值,再根据样本的各维数据值和所得到的对应权值进行三次样条插值,得到各维数据值的权值函数,而不是传统方法的常数,这克服了个别数据变化所带来的整体度量值发生较大变化的缺点.数值仿真实验表明:分维权重样条插值预测算法不失是一种稳定而灵活的算法,而且预测的精度较高,可以根据样条插值函数得到样本各维的权值.     

Quadratic spline interpolation with coinciding interpolation and spline grids

In this paper the quadratic spline interpolation with coinciding interpolation and spline grids for continuous functions is considered. The theorems mainly concern error estimations which allow to formulate a convergence statement. To get such results it is assumed that the function to be interpolated is suitably smooth or possesses a special behavior. A best approximation property and a statement about the solution of boundary value problems using quadratic spline functions are added.