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.     

Natural bicubic spline fractal interpolation

Fractal Interpolation functions provide natural deterministic approximation of complex phenomena. Cardinal cubic splines are developed through moments (i.e. second derivative of the original function at mesh points). Using tensor product, bicubic spline fractal interpolants are constructed that successfully generalize classical natural bicubic splines. An upper bound of the difference between the natural cubic spline blended fractal interpolant and the original function is deduced. In addition, the convergence of natural bicubic fractal interpolation functions towards the original function providing the data is studied.     

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…     

An algorithm for least-squares fitting of cubic spline surfaces to functions on a rectilinear mesh over a rectangle

In this paper a method is presented for fitting, in the least-squares sense, a bivariate cubic spline function to values of a dependent variable, specified at points on a rectangular grid in the plane of the independent variables. Products of B-splines are used to represent the bicubic splines. The coefficients in this representation are determined by solving a set of one-dimensional least-squares problems.     

(3,3,1,1)层次B样条与分级T网格上多项式样条的相互转换

1引言 B样条在计算机图形学和几何建模等领域有着广泛的应用[3,8].在应用过程中,通常都需要对得到的模型进行修改以到达更好的效果.对于B样条曲线,利用节点插入算法可以有效地进行局部修改.     

An approximation problem of noisy data by cubic and bicubic splines

Smoothing of noisy data has always been a topic of interest in many areas where computer simulations have been performed, including natural sciences as well as economics and social sciences. In this paper we present an approximation method of explicit curves or surfaces from exact and noisy data by fairness cubic or bicubic splines. A variational problem of explicit curves or surfaces is obtained by minimizing a quadratic functional in a space of cubic or bicubic splines from noisy data. We show the existence and uniqueness of this problem as long as a convergence result especially for noisy data is carefully established. We analyze some numerical and graphical examples using fictional noisy data in order to prove the validity of our method.     

多变量样条元法分析弹性地基板的弯曲,振动与稳定问题

本文应用双三次乘积型二元B样条函数来构造弹性地基板的位移、弯矩和扭矩等多种场函数，由混合变分原理导出多变量样条无法方程．文中，对弹性地基板的弯曲、振动与稳定问题作了分析与计算．由于，本文方法设定了各自独立的场函数，因此，所算得的场未知量如位移、弯矩和扭矩值的精度均比较高。     

S-convex,monotone, and positive interpolation with rational bicubic splines of C2-continuity

In this paper we deal with shape preserving interpolation of data sets given on rectangular grids. The aim is to show that there exist spline interpolants of the continuity classC ² which areS-convex, monotone, or positive if the data sets have these properties. This is done by using particular rational bicubic splines defined on the grids introduced by the data. Interpolants of the desired type can be constructed by a simple search procedure.     

Numerical Procedure for Calibration of Volatility with American Options

In finance, the price of an American option is obtained from the price of the underlying asset by solving a parabolic variational inequality. The calibration of volatility from the prices of a family of American options yields an inverse problem involving the solution of the previously mentioned parabolic variational inequality. In this paper, the discretization of the variational inequality by finite elements is studied in detail. Then, a calibration procedure, where the volatility belongs to a finite‐dimensional space (finite element or bicubic splines) is described. A least square method, with suitable regularization terms is used. Necessary optimality conditions involving adjoint states are given and the differentiability of the cost function is studied. A parallel algorithm is proposed and numerical experiments, on both academic and realistic cases, are presented.     

Estimation of risk-neutral density surfaces

Option price data is often used to infer risk-neutral densities for future prices of an underlying asset. Given the prices of a set of options on the same underlying asset with different strikes and maturities, we propose a nonparametric approach for estimating risk-neutral densities associated with several maturities. Our method uses bicubic splines in order to achieve the desired smoothness for the estimation and an optimization model to choose the spline functions that best fit the price data. Semidefinite programming is employed to guarantee the nonnegativity of the densities. We illustrate the process using synthetic option price data generated using log-normal and absolute diffusion processes as well as actual price data for options on the S&P 500 index. We also used the risk-neutral densities that we computed to price exotic options and observed that this approach generates prices that closely approximate the market prices of these options.     

Local cubic splines on non-uniform grids and real-time computation of wavelet transform

In this paper, local cubic quasi-interpolating splines on non-uniform grids are described. The splines are designed by fast computational algorithms that utilize the relation between splines and cubic interpolation polynomials. These splines provide an efficient tool for real-time signal processing. As an input, the splines use either clean or noised arbitrarily-spaced samples. Formulas for the spline’s extrapolation beyond the sampling interval are established. Sharp estimations of the approximation errors are presented. The capability to adapt the grid to the structure of an object and to have minimal requirements to the operating memory are of great advantages for offline processing of signals and multidimensional data arrays. The designed splines serve as a source for generating real-time wavelet transforms to apply to signals in scenarios where the signal’s samples subsequently arrive one after the other at random times. The wavelet transforms are executed by six-tap weighted moving averages of the signal’s samples without delay. On arrival of new samples, only a couple of adjacent transform coefficients are updated in a way that no boundary effects arise.     

任意剖分下的多元样条分析

本文采用代数几何的方法,研究了在任意剖分下多元样条函数的各种性质.定理2—4给出了一个函数S(υ,ν)是多元参数型样条的充分必要条件.定理1指出了多元样条函数具有“解析延拓”的特征性质.文中得到在任意剖分下多元样条的一般表达形式(定理9和10)和多元样条插值的一般理论.文中也讨论了多元有理样条函数.     

Spline Surfaces over Arbitrary Topological Meshes: Theoretical Analysis and Application

Based on polyhedral splines, some multivariate splines of different orders with given supports over arbitrary topological meshes are developed. Schemes for choosing suitable families of multivariate splines based on pre-given meshes are discussed. Those multivariate splines with inner knots and boundary knots from the related meshes are used to generate rational spline shapes with related control points. Steps for up to $C^2$-surfaces over the meshes are designed. The relationship among the meshes and their knots, the splines and control points is analyzed. To avoid any unexpected discontinuities and get higher smoothness, a heart-repairing technique to adjust inner knots in the multivariate splines is designed.With the theory above, bivariate $C^1$-quadratic splines over rectangular meshes are developed. Those bivariate splines are used to generate rational $C^1$-quadratic surfaces over the meshes with related control points and weights. The properties of the surfaces are analyzed. The boundary curves and the corner points and tangent planes, and smooth connecting conditions of different patches are presented. The $C^1$−continuous connection schemes between two patches of the surfaces are presented.     

The complexity of numerical methods for elliptic partial differential equations

We consider three Ritz-Galerkin procedures with Hermite bicubic, bicubic spline and linear triangular elements for approximating the solution of self-adjoint elliptic partial differential equations and a collocation with Hermite bicubics method for general linear elliptic equations defined on general two dimensional domains with mixed boundary conditions. We systematically evaluate these methods by applying them to a sample set of problems while measuring various performance criteria. The test data suggest that collocation is the most efficient method for general use.     

一种基于函数值的二元有理插值函数及其性质

利用带参数的仅以被插函数的函数值作为插值条件的一元有理插值方法,构造了一种分母为双二次的仪基于函数值的二元有理双三次插值函数,插值函数具有简洁的显示表示,插值函数中含有四个参数,当这些参数满足一定条件时,插值曲而在插值区域上C1光滑.由于插值函数中含有参数,这样可以在插值数据不变的情况下通过对参数的选择进行插值曲面的局部修改,最后讨论了插值函数的一些性质.     

Statistical application of barycentric rational interpolants: an alternative to splines

Spline curves, originally developed by numerical analysts for interpolation, are widely used in statistical work, mainly as regression splines and smoothing splines. Barycentric rational interpolants have recently been developed by numerical analysts, but have yet seen very few statistical applications. We give the necesssary information to enable the reader to use barycentric rational interpolants, including a suggestion for a Bayesian prior distribution, and explore the possible statistical use of barycentric interpolants as an alternative to splines. We give the all the necessary formulae, compare the numerical accuracy to splines for some Monte-Carlo datasets, and apply both regression splines and barycentric interpolants to two real datasets. We also discuss the application of these interpolants to data smoothing, where smoothing splines would normally be used, and exemplify the use of smoothing interpolants with another real dataset. Our conclusion is that barycentric interpolants are as accurate as splines, and no more difficult to understand and program. They offer a viable alternative methodology.     

Complex Wavelets and Framelets from Pseudo Splines

In this paper, we introduce complex pseudo splines that are derived from pseudo splines of type I. First, we show that the shifts of every complex pseudo spline are linearly independent. Therefore we can construct a biorthogonal wavelet system. Next, we investigate the Riesz basis property of the corresponding wavelet system generated by complex pseudo splines. The regularity of the complex pseudo splines will be analyzed. Furthermore, by using complex pseudo splines, we will construct symmetric or antisymmetric complex tight framelets with desired approximation order.     

The Structural Characterization and Locally Supported Bases for Bivariate Super Splines

Super splines are bivariate splines defined on triangulations, where the smoothness enforced at the vertices is larger than the smoothness enforced across the edges. In this paper, the smoothness conditions and conformality conditions for super splines are presented. Three locally supported super splines on type-1 triangulation are presented. Moreover, the criteria to select local bases is also givenBy using local supported super spline function, a variation-diminishing operator is built. The approximation properties of the operator are also presented.     

双三次B样条插值曲面

本文研究三次B样条插值曲面。对于给定的拓扑网格点阵Pi,j。导出了其插值三次B样条曲面的控制顶点,每四个顶点Pi,j,Pi l,j,Pi,j 1,Pi 1,j 1由九个三次B样条曲面片构成,整个曲面是C^2连续的,最后,给出了一个数值实例。     

Some recurrence formulas for box splines and cone splines

A degree elevation formula for multivariate simplex splines was given by Micchelli [6] and extended to hold for multivariate Dirichlet splines in [8]. We report similar formulae for multivariate cone splines and box_splines. To this end, we utilize a relation due to Dahmen and Micchelli [4] that connects box splines and cone splines and a degree reduction formula given by Cohen, Lyche, and Riesenfeld in [2].