Shepard Operator Research Articles

A kind of even order Bernoulli-type operator with bivariate Shepard

<abstract><p>It is known that an efficient method for interpolation of very large scattered data sets is the method of Shepard. Unfortunately, it reproduces only the constants. In this paper, we first generalize an expansion in bivariate even order Bernoulli polynomials for real functions possessing a sufficient number of derivatives. Finally, by combining the known Shepard operator with the even order Bernoulli bivariate operator, we construct a kind of new approximated operator satisfying the higher order polynomial reproducibility. We study this combined operator and give some error bounds in terms of the modulus of continuity of high order and also with Peano's theorem. Numerical comparisons show that this new technique provides the higher degree of accuracy. Furthermore, the advantage of our method is that the algorithm is very simple and easy to implement.</p></abstract>

On the improvement of the triangular Shepard method by non conformal polynomial elements

In this paper, we introduce a new nonconforming finite element as a polynomial enrichment of the standard triangular linear element. Based on this new element, we propose an improvement of the triangular Shepard operator. We prove that the order of this new approximation operator is at least cubic. Numerical experiments demonstrate the accuracy of the proposed method.

Approximation by Kantorovich-type max-min operators and its applications

In this study, we construct Kantorovich variant of max-min kind operators, which are nonlinear. By using these new operators, we obtain some uniform approximation results in N-dimension (N≥1). Then, we estimate the error with the help of Hölder continuous functions and modulus of continuity. Furthermore, we give some illustrative applications to verify our theory and also investigate some shape-preserving properties of Kantorovich-type max-min Bernstein operator. Lastly, we examine the image processing implementation of our results via Kantorovich-type max-min Shepard operator.

The combined Shepard operator of inverse quadratic and inverse multiquadric type

"Starting with the classical, the modified and the iterative Shepard methods, we construct some new Shepard type operators, using the inverse quadratic and the inverse multiquadric radial basis functions. Given some sets of points, we compute some representative subsets of knot points following an algorithm described by J.R. McMahon in 1986."

Solving Poisson equation with Dirichlet conditions through multinode Shepard operators

The multinode Shepard operator is a linear combination of local polynomial interpolants with inverse distance weighting basis functions. This operator can be rewritten as a blend of function values with cardinal basis functions, which are a combination of the inverse distance weighting basis functions with multivariate Lagrange fundamental polynomials. The key for simply computing the latter, on a unisolvent set of points, is to use a translation of the canonical polynomial basis and the PA=LU factorization of the associated Vandermonde matrix. In this paper, we propose a method to numerically solve a Poisson equation with Dirichlet conditions through multinode Shepard interpolants by collocation. This collocation method gives rise to a collocation matrix with many zero entrances and a smaller condition number with respect to the one of the well known Kansa method. Numerical experiments show the accuracy and the performance of the proposed collocation method.

Shepard operator of least squares thin-plate spline type

"We obtain some new Shepard type operators based on the classical, the modi ed Shepard methods and the least squares thin-plate spline function. Given some sets of points, we compute some representative subsets of knot points following an algorithm described by J. R. McMahon in 1986."

On the hexagonal Shepard method

The problem of Lagrange interpolation of functions of two variables by quadratic polynomials based on nodes which are vertices of a triangulation has been recently studied and local six-tuples of vertices which assure the uniqueness and the optimal-order of the interpolation polynomial are known. Following the idea of Little and the theoretical results on the approximation order and accuracy of the triangular Shepard method, we introduce an hexagonal Shepard operator with quadratic precision and cubic approximation order for the classical problem of scattered data approximation without least square fit.

Increasing the approximation order of the triangular Shepard method

In this paper we discuss an improvement of the triangular Shepard operator proposed by Little to extend the Shepard method. In particular, we use triangle based basis functions in combination with a modified version of the linear local interpolant on the vertices of the triangle. We deeply study the resulting operator, which uses functional and derivative data, has cubic approximation order and a good accuracy of approximation. Suggestions on how to avoid the use of derivative data, without losing both order and accuracy of approximation, are given.

An Iterative Modification of Shepard–Bernoulli Operator

We introduce an iterative modification of the combined Shepard operator of Bernoulli type, introduced in Cătinas (Calcolo 44:189–202, 2007), that is free from artificial setup of parameters, and corresponds to successive scaling.

On the approximation order of triangular Shepard interpolation

Shepard’s method is a well-known technique for interpolating large sets of scattered data. The classical Shepard operator reconstructs an unknown function as a normalized blend of the function values at the scattered points, using the inverse distances to the scattered points as weight functions. Based on the general idea of defining interpolants by convex combinations, Little suggests to extend the bivariate Shepard operator in two ways. On the one hand, he considers a triangulation of the scattered points and substitutes function values with linear polynomials which locally interpolate the given data at the vertices of each triangle. On the other hand, he modifies the classical point-based weight functions and defines instead a normalized blend of the locally interpolating polynomials with triangle-based weight functions which depend on the product of inverse distances to the three vertices of the corresponding triangle. The resulting triangular Shepard operator interpolates all data required for its definition and reproduces polynomials up to degree 1, whereas the classical Shepard operator reproduces only constants. In this paper we show that this interpolation operator consequentially has quadratic approximation order, which is confirmed by our numerical results. Citation Info Journal IMA Journal of Numerical Analysis Volume 36(1), January 2016 Pages 359–379

Scattered data quasi‐interpolation on spheres

This paper studies the construction and approximation of quasi‐interpolation for spherical scattered data. First of all, a kind of quasi‐interpolation operator with Gaussian kernel is constructed to approximate the spherical function, and two Jackson type theorems are established. Second, the classical Shepard operator is extended from Euclidean space to the unit sphere, and the error of approximation by the spherical Shepard operator is estimated. Finally, the compact supported kernel is used to construct quasi‐interpolation operator for fitting spherical scattered data, where the spherical modulus of continuity and separation distance of scattered sampling points are employed as the measurements of approximation error. Copyright © 2014 John Wiley & Sons, Ltd.

Finding better weight functions for generalized Shepard's operator on infinite intervals

In this paper, we investigate the difference of Shepard's generalized operators S σ from the approximated set of data for various weight functions σ. Bounds are given for the sizes of the ‘bumps’ shown on the graph of S σ for σ(d)=1/d in dimension N=1, and the best weight function σ for practical use is proposed.

The bivariate Shepard operator of Bernoulli type

The method of Shepard is an efficient method for interpolation of very large scattered data sets; unfortunately, it has poor reproduction qualities and high computational cost. In this paper we introduce a new operator which diminishes these drawbacks. This operator results from the combination of the Shepard operator with a new interpolation operator, recently proposed by Costabile and Dell’Accio, and generalizes to two variate functions the Shepard-Bernoulli operator introduced in [2]. We study this combined operator and give error bounds in terms of the modulus of continuity of high order and of the mesh length. We improve the accuracy and computational efficiency using a method introduced by Franke and Nielson. Keywords: Shepard operator, Bernoulli operator, interpolation of scattered data, error estimations. Mathematics Subject Classification (2000): 41A05, 41A25, 41A80.

Shepard--Bernoulli operators

We introduce the Shepard-Bernoulli operator as a combination of the Shepard operator with a new univariate interpolation operator: the generalized Taylor polynomial. Some properties and the rate of convergence of the new combined operator are studied and compared with those given for classical combined Shepard operators. An application to the interpolation of discrete solutions of initial value problems is given.

The combined Shepard-Abel-Goncharov univariate operator

We extend the Shepard operator by combining it with the Abel-Goncharov univariate operator in order to increase the degree of exactness and to use some specific functionals. We study this combined operator and give some of its properties. We introduce the corresponding interpolation formula and study its remainder term.

Notes on the approximation rate of fuzzy KH interpolators

This paper investigates the approximation behaviour of the Kóczy–Hirota (KH) interpolative fuzzy controllers. First, in accordance with the remarks in (Fuzzy Sets and Systems 125(1) (2002) 105), it is pointed out that it is a fuzzy generalization of the Shepard operator. Shepard operator has thoroughly studied by approximation theorist since the mid-1970s. Exploiting the aforementioned relationship, we establish analog results on the approximation rate of KH controllers. The optimal order and class of approximation (saturation problem) are determined for certain values of the exponent λ. Corresponding results on the modified alpha-cut based interpolation method, being an improvement of the KH interpolator, are also provided. The results offer trade-off facilities between approximation accuracy and the number of rules. As a consequence, the necessary and sufficient number of rules can be determined for a prescribed accuracy.

Balázs–Shepard Operators on Infinite Intervals, II

The weighted uniform convergence of the Balázs–Shepard operator is considered on the real line. As a consequence of the main result, it is proved that for a wide class of weights, rational functions are always dense in the space of continuous functions, in contrast to the polynomials where the Akhiezer–Babenko condition is necessary for such density.

Saturation of the shepard operator

Saturation of the shepard operator

Direct and converse approximation theorems for the Shepard operator

Direct and converse approximation theorems for the Shepard operator (1) are given in uniform metric. The main result is Theorem 3 which completes the characterization of Lipschitz classes by the order of approximation by the Shepard operator for λ>2.