Construction of asymptotically best quadrature formulas

Summary. We show that, for integrals with arbitrary integrable weight functions, asymptotically best quadrature formulas with equidistant nodes can be obtained by applying a certain scheme of piecewise polynomial interpolation to the function to be integrated, and then integrating this interpolant. Received August 7, 1991     

A class of piecewise interpolating functions based on barycentric coordinates

Piecewise interpolation methods, as spline or Hermite cubic interpolation methods, define the interpolant function by means of polynomial pieces and ensure that some regularity conditions are guaranteed at the break-points. In this work, we propose a novel class of piecewise interpolating functions whose expression depends on the barycentric coordinates and a suitable weight function. The underlying idea is to specialize to the 1D settings some aspects of techniques widely used in multi-dimensional interpolation, namely Shepard’s, barycentric and triangle-based blending methods. We show the properties of convergence for the interpolating functions and discuss how, in some cases, the properties of regularity that characterize the weight function are reflected on the interpolant function. Numerical experiments, applied to some case studies and real scenarios, show the benefit of our method compared to other interpolant models.     

Shape-Preserving Local Interpolation for Plotting Solutions of ODEs

We investigate the use of piecewise rational interpolants ofDelbourgo and Gregory in an important and widely occurring application.We propose the following algorithm for visually pleasing plotsof the solution of an ordinary differential equation (ODE):use piecewise cubic Hermite interpolation where it can be shownto preserve shape (monotonicity and/or convexity) and also wherethere is no shape to preserve, otherwise use the appropriateconvex or monotone piecewise rational interpolant. Bounds arederived which enable efficient plotting of the rational interpolants.This scheme should be useful in any context where both solutionand derivative of a function are available as data.     

Error Bound for Bernstein-Bézier Triangular Approximation

Based upon a new error bound for the linear interpolant to a function defined on a triangle and having continuous partial derivatives of second order, the related error bound for n-th Bernstein triangular approximation is obtained. The order of approximation is 1/n.     

Scattered data interpolation using minimum energy Powell-Sabin elements and data dependent triangulations

A popular approach for obtaining surfaces interpolating to scattered data is to define the interpolant in a piecewise manner over a triangulation with vertices at the data points. In most cases, the interpolant cannot be uniquely determined from the prescribed function values since it belongs to a space of functions of dimension greater than the number of data points. Thus, additional parameters are needed to define an interpolant and have to be estimated somehow from the available data. It is intuitively clear that the quality of approximation by the interpolant depends on the choice of the triangulation and on the method used to provide the additional parameters. In this paper we suggest basing the selection of the triangulation and the computation of the additional parameters on the idea of minimizing a given cost functional measuring the quality of the interpolant. We present a scheme that iteratively updates the triangulation and computes values of the additional parameters so that the quality of the interpolant, as measured by the cost functional, improves from iteration to iteration. This method is discussed and tested numerically using an energy functional and Powell-Sabin twelve split interpolants.     

On Shape Preserving C2 Hermite Interpolation

We propose a general parametric local approach for functional C ² Hermite shape preserving interpolation. The constructed interpolant is a parametric curve which interpolate values, first and second derivatives of a given function and reproduces the behavior of the data. The method is detailed for parametric curves with piecewise cubic components. For the selected space necessary and sufficient conditions are derived to ensure the convexity of the constructed interpolant. Monotonicity is also studied. The approximation order is investigated for both cases. The use of a parametric curves to interpolate data from a function can be considered a disadvantage of the scheme. However, the simple structure of the used curve greatly reduces such a disadvantage.     

Comonotone parametric C1 interpolation of nongridded data

We present a local scheme for constructing a C¹ interpolating function comonotone with a set of data having a tensor product topology. The constructed interpolant is a parametric surface with piecewise bicubic components which locally maintain the monotonicity of the data along curves “parallel” to the related coordinate lines. Error estimates and graphical examples are provided.     

Piecewise monotone interpolation

In reaction to a recent paper by E. Passow in this Journal, it is shown that broken line interpolation as a scheme for piecewise monotone interpolation is hard to improve upon. It is also shown that a family of smooth piecewise polynomial interpolants, introduced by Swartz and Varga and noted by Passow to be piecewise monotone, converges monotonely, for fixed data, to a piecewise constant interpolant as the degree goes to infinity. Finally, piecewise monotone interpolation by splines with simple knots is discussed.     

Polynomial of Degree Four Interpolation on Triangles

A method for constructing the $C^1$ piecewise polynomial surface of degree four on triangles is presented in this paper. On every triangle, only nine interpolation conditions, which are function values and first partial derivatives at the vertices of the triangle, are needed for constructing the surface.     

单调光滑函数的样条插值

ELI Passow在[1]中提出了X={x_i}(x_(i-1)相似文献   

有理线性插值曲线和曲面

构造了含参数的分段线性有理插值函数(分子、分母均为一次多项式),通过适当选择形状参数,由此函数产生的曲线一阶连续并且保单调.文中用张量积方法将此结果推广到二元矩形网格上的曲面插值,同时给出了插值函数的误差估计及数值例子.     

Piecewise Linear Interpolation With Nonequidistant Nodes

Pointwise and norm estimates for approximation of continuous function by piecewise linear interpolant with nonequidistant nodes are proved. Special attention is paid to the value of the absolute constants in front of the modulus.     

A multivariate Powell–Sabin interpolant

We consider the problem of constructing a C ¹ piecewise quadratic interpolant, Q, to positional and gradient data defined at the vertices of a tessellation of n-simplices in \mathbbRⁿ \mathbb{R}^{n} . The key to the interpolation scheme is to appropriately subdivide each simplex to ensure that certain necessary geometric constraints are satisfied by the subdivision points. We establish these constraints using the Bernstein–Bézier form for polynomials defined over simplices, and show how they can be satisfied. When constructed, the interpolant Q has full approximation power.     

参数保形插值中决定切矢的方法

由分段三次参数多项式曲线拼合成的Ｃ１插值曲线的形状与数据点处的切矢有很大关系．基于对保形插值曲线特点的分析，本文提出了估计数据点处切矢的一种方法：采用使构造的插值曲线的长度尽可能短的思想估计数据点处的切矢，并且通过四组有代表性的数据对本方法和已有的三种方法进行了比较．     

C1 monotone cubic Hermite interpolant

Constraining an interpolation to be shape preserving is a well established technique for modelling scientific data. Many techniques express the constraint variables in terms of abstract quantities that are difficult to relate to either physical values or the geometric properties of the interpolant. In this paper, we construct a piecewise monotonic interpolant where the degrees of freedom are expressed in terms of the weights of the rational Bézier cubic interpolant.     

Piecewise Rational Quadratic Interpolation to Monotonic Data

An explicit representation of a piecewise rational quadraticfunction is developed which produces a monotonic interpolantto given monotonic data. The explicit representation means thatthe piecewise monotonic interpolant is easily constructed andnumerical experiments indicate that the method produces visuallypleasing curves. Furthermore, the use of the method is justifiedby a convergence analysis.     

一种二次保形插值参数曲面

1．引言保形插值是工业设计和制造中经常遇到的问题，有关这方面的研究已有许多文献【‘－u1．设n二Fx；，yi，人，川7一0，1，…；n；j＝0；l，…，。；x；＜x；＋1．l＝0，1，…．n—1；yi＜的十；，J二O，L…，。一卫｝是一给定的数据集，Cadson和［ltsch、Beatson和zejerJ－1985年分别提出的方法只保持被插数据集的轴向单调性；Dodd和Roulier等人于1983和1987年提出的方法只保持被插点集网格线上的轴向凸凹性和单调性；Constantini和FOntanella于1990年提出的方法可保持被插点集在所有于区域的边界及共内部的轴向凸凹性和单调性；…     

Admissible slopes for monotone and convex interpolation

Summary In many applications, interpolation of experimental data exhibiting some geometric property such as nonnegativity, monotonicity or convexity is unacceptable unless the interpolant reflects these characteristics. This paper identifies admissible slopes at data points of variousC ¹ interpolants which ensure a desirable shape. We discuss this question, in turn for the following function classes commonly used for shape preserving interpolations: monotone polynomials,C ¹ monotone piecewise polynomials, convex polynomials, parametric cubic curves and rational functions.     

保凸C^1分段二次多项式插值方法

本导出了二次多项式保凸的充要条件,通过插值部分新节点,得到了一种新的保凸Ｃ＾１分段二镒多项式插值函数。     

High-resolution difference methods with exact evolution for multidimensional waves

We consider the generalization of high-order upwind Strang methods for simulating waves. In $1+1$ dimensions the methods can be defined via the exact evolution over a single time step of an odd-order piecewise polynomial interpolant of the grid data. We construct a true multidimensional version for acoustic waves by applying the solution operator in integral form to the interpolant. We also examine the replacement of polynomials by bandlimited interpolation functions (BLIFs). Numerical experiments with turbulent wave fields are presented to verify the accuracy and stability of the multidimensional methods and to assess the relative effectiveness of the two interpolation techniques.