C2-rational cubic spline involving tension parameters

In the present paper, C¹-piecewise rational cubic spline function involving tension parameters is considered which produces a monotonie interpolant to a given monotonie data set. It is observed that under certain conditions the interpolant preserves the convexity property of the data set. The existence and uniqueness of a C²-rational cubic spline interpolant are established. The error analysis of the spline interpolant is also given.     

关于n维单形保多项式超限插值的表示问题

以R~n表示n维欧氏空间,Z_+~n是R~n中坐标均为非负整数的全体,e~s为Z_+~(n+1)中第s个坐标为1其余坐标为0的单位向量;π_d(R~n)为全次数不大于d的n元多项式全体,     

A nodal spline interpolant for the Gregory rule of even order

Summary The Gregory rule is a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order. In the literature, the methods of constructing the Gregory rule have, in contrast to Newton-Cotes quadrature,not been based on the integration of an interpolant. In this paper, after first characterizing an even-order Gregory interpolant by means of a generalized Lagrange interpolation operator, we proceed to explicitly construct such an interpolant by employing results from nodal spline interpolation, as established in recent work by the author and C.H. Rohwer. Nonoptimal order error estimates for the Gregory rule of even order are then easily obtained.     

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.     

Optimal methods of interpolation in Nonparametric Regression

Within the framework of Optimal Recovery, optimal methods of interpolation, based on the Abel–Jacobi elliptic functions, have been found for some Hardy classes of analytic functions [9]. It will be shown that these methods are also optimal according to criteria of Optimal Design and Nonparametric Regression.For all noise levels away from 0, the mean squared error of the optimal interpolant is evaluated explicitly, in a non-asymptotic setting. In this result, a pivotal role is played by an interference effect in which both stochastic and deterministic parts of the interpolant exhibit an oscillating behavior, with the two oscillating functions canceling each other.     

Favard’s interpolation problem in one or more variables

Given scattered data on the real line, Favard [4] constructed an interpolant which depends linearly and locally on the data and whose nth derivative is locally bounded by the nth divided differences of the data times a constant depending only on n. It is shown that the (n —1)th derivative of Favard’s interpolant can be likewise bounded by divided differences, and that one can bound at best two consecutive derivatives of any interpolant by the corresponding divided differences. In this sense, Favard’s univariate interpolant is the best possible. Favard’s result has been extended [8] to a special case in several variables, and here the extent to which this can be repeated in a more general setting is proven exactly.     

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.     

Transfinite mean value interpolation in general dimension

Mean value interpolation is a simple, fast, linearly precise method of smoothly interpolating a function given on the boundary of a domain. For planar domains, several properties of the interpolant were established in a recent paper by Dyken and the second author, including: sufficient conditions on the boundary to guarantee interpolation for continuous data; a formula for the normal derivative at the boundary; and the construction of a Hermite interpolant when normal derivative data is also available. In this paper we generalize these results to domains in arbitrary dimension.     

Rational interpolation with a single variable pole

In this paper the necessary and sufficient conditions for given data to admit a rational interpolant in _k,1 with no poles in the convex hull of the interpolation points is studied. A method for computing the interpolant is also provided.Partially supported by DGICYT-0121.     

ON TRIANGULAR C~1 SCHEMES: A NOVEL CONSTRUCTION

1. IntroductionThe smooth interpolation on a triangulation of a planar region is of great importancein most applied areas) such as computation of finite element method, computer aided(geometric) design and scattered data processing.Let A be a triangulation of a polygonal domain fi C RZ and Ac, al and aZ the setso f venices, edges and triangles in a respectively. Usually the triangulation in practiceis formed by a mass of scattered nodes that, covered by the region fi, are carryingsimilar typ…     

On the structure of a table of multivariate rational interpolants

In the table of multivariate rational interpolants the entries are arranged such that the row index indicates the number of numerator coefficients and the column index the number of denominator coefficients. If the homogeneous system of linear equations defining the denominator coefficients has maximal rank, then the rational interpolant can be represented as a quotient of determinants. If this system has a rank deficiency, then we identify the rational interpolant with another element from the table using less interpolation conditions for its computation and we describe the effect this dependence of interpolation conditions has on the structure of the table of multivariate rational interpolants. In the univariate case the table of solutions to the rational interpolation problem is composed of triangles of so-called minimal solutions, having minimal degree in numerator and denominator and using a minimal number of interpolation conditions to determine the solution.Communicated by Dietrich Braess.     

A model-trust region algorithm utilizing a quadratic interpolant

A new model-trust region algorithm for problems in unconstrained optimization and nonlinear equations utilizing a quadratic interpolant for step selection is presented and analyzed. This is offered as an alternative to the piecewise-linear interpolant employed in the widely used “double dogleg” step selection strategy. After the new step selection algorithm has been presented, we offer a summary, with proofs, of its desirable mathematical properties. Numerical results illustrating the efficacy of this new approach are presented.     

Shape Preserving Interpolation to Convex Data

1.IntroductionA fundamental problem in computer graphics is the drawing of a smooth curve through aset of data points(xi,fi) (i=0 ,1 ,… ,n) .In many applications,particularly in scientificvisualisation,the y- values are depenenton the x- values and it is…     

On the Semi-norm of Radial Basis Function Interpolants

Radial basis function interpolation has attracted a lot of interest in recent years. For popular choices, for example thin plate splines, this problem has a variational formulation, i.e. the interpolant minimizes a semi-norm on a certain space of radial functions. This gives rise to a function space, called the native space. Every function in this space has the property that the semi-norm of an arbitrary interpolant to this function is uniformly bounded. In applications it is of interest whether a sufficiently smooth function belongs to the native space. In this paper we give sufficient conditions on the differentiability of a function with compact support, in the case of cubic, linear and thin plate splines. In the case of multiquadrics and Gaussian functions, it is shown that the only compactly supported function that satisfies these conditions is identically zero.     

Towards the analysis of materials with strain gradient effects using -Natural Element Method

The notion of generalized continua unifies several classes of continuum theories that account for size dependence arising due to the underlying microstructure of the material. In gradient continua, besides the first gradient also the higher gradients of displacements are taken in to account. This results in higher order boundary conditions and C¹ continuity requirements. α-Natural Element Method has been used to this effect for numerical modeling of gradient elasticity. The α-–Natural Element Method allows the construction of models entirely in terms of nodes and also ensures the linear precision of the interpolant over convex and non convex boundaries. C¹ natural neighbor interpolants are achieved by a simple transformation of the Farins interpolant, which are basically obtained by embedding Sibsons natural neighbor coordinates in Bernstein–Bezier surface representation of a cubic simplex. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transfinite Thin Plate Spline Interpolation

Duchon’s method of thin plate splines defines a polyharmonic interpolant to scattered data values as the minimizer of a certain integral functional. For transfinite interpolation, i.e., interpolation of continuous data prescribed on curves or hypersurfaces, Kounchev has developed the method of polysplines, which are piecewise polyharmonic functions of fixed smoothness across the given hypersurfaces and satisfy some boundary conditions. Recently, Bejancu has introduced boundary conditions of Beppo–Levi type to construct a semicardinal model for polyspline interpolation to data on an infinite set of parallel hyperplanes. The present paper proves that, for periodic data on a finite set of parallel hyperplanes, the polyspline interpolant satisfying Beppo–Levi boundary conditions is in fact a thin plate spline, i.e., it minimizes a Duchon type functional. The construction and variational characterization of the Beppo–Levi polysplines are based on the analysis of a new class of univariate exponential ℒ-splines satisfying adjoint natural end conditions.     

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.     

Multivariate convexity preserving interpolation by smooth functions

A set of multivariate data is called strictly convex if there exists a strictly convex interpolant to these data. In this paper we characterize strict convexity of Lagrange and Hermite multivariate data by a simple property and show that for strict convex data and given smoothness requirements there exists a smooth strictly convex interpolant. We also show how to construct a multivariate convex smooth interpolant to scattered data. Communicated by T.N.T. Goodman     

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.     

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．