On the bivariate Shepard–Lidstone operators

Caira, R.; Dell’Accio, Francesco; Tommaso, Filomena Di

doi:10.1016/j.cam.2011.10.001

Cited by 28 publications

(11 citation statements)

References 14 publications

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“…This drawback can be avoided by considering various methods; for instance, it is possible to use partition of unity methods [24,25] or Shepard's like methods [26][27][28][29][30][31][32]. This problem can also be avoided by using B-spline quasi-interpolation.…”

Section: New Quasi-interpolation Methodsmentioning

confidence: 99%

A New Approximation Method with High Order Accuracy

Jiang

2017

MCA

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Abstract:In this paper, we propose a new multilevel univariate approximation method with high order accuracy using radial basis function interpolation and cubic B-spline quasi-interpolation. The proposed approach includes two schemes, which are based on radial basis function interpolation with less center points, and cubic B-spline quasi-interpolation operator. Error analysis shows that our method produces higher accuracy compared with other approaches. Numerical examples demonstrate that the proposed scheme is effective.

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Section: New Quasi-interpolation Methodsmentioning

confidence: 99%

A New Approximation Method with High Order Accuracy

Jiang

2017

MCA

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“…As a consequence, the resulting operator not only inherits interpolation conditions that each three point local interpolation polynomial satisfies at the referring vertex and increases by 1 the degree of exactness of the Shepard-Taylor operator [21] which uses the same data, but also improves its accuracy. In this sense, the Shepard-Bernoulli operators belong to a recently introduced class of operators for enhancing the approximation order of Shepard operators by using supplementary derivative data [7,15,16]. For the general problem of the enhancement of the algebraic precision of linear operators of approximation see the papers [26,25,30] and the references therein.…”

Section: Discussionmentioning

confidence: 99%

Bivariate Shepard–Bernoulli operators

Dell’Accio

Tommaso

2017

Mathematics and Computers in Simulation

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Abstract. In this paper we extend the Shepard-Bernoulli operators introduced in [6] to the bivariate case. These new interpolation operators are realized by using local support basis functions introduced in [23] instead of classical Shepard basis functions and the bivariate three point extension [13] of the generalized Taylor polynomial introduced by F. Costabile in [11]. The new operators do not require either the use of special partitions of the node convex hull or special structured data as in [8]. We deeply study their approximation properties and provide an application to the scattered data interpolation problem; the numerical results show that this new approach is comparable with the other well known bivariate schemes QSHEP2D and CSHEP2D by Renka [34,35]. The problemLet N = {x 1 , ..., x N } be a set of N distinct points (called nodes or sample points) of R s , s ∈ N, and let f be a function defined on a domain D containing N . The classical Shepard operators (first introduced in [37] in the particular case s = 2) are defined bywhere the weight functions A µ,i (x) in barycentric form areand |·| denotes the Euclidean norm in R s . The interpolation operator S N,µ [·] is stable, in the sense thatbut for µ > 1 the interpolating function S N,µ [f ] (x) has flat spots in the neighborhood of all data points. Moreover, the degree of exactness of the operator S N,µ [·] is 0, i.e. if it is restricted to the polynomial space P

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“…The latter are then used in combination with linear polynomials that locally interpolate the given data at the vertices of each triangle. Polynomials based on the vertices of a triangle [4,5,6] are used in combination with Shepard basis functions in [7,8,9,10]. To extend the point-based basis functions in (2) to triangle-based basis functions, let us consider a triangulation…”

Section: Shepard and Triangular Shepard Operatorsmentioning

confidence: 99%

On the enhancement of the approximation order of triangular Shepard method

Dell’Accio¹,

Tommaso²,

Hormann³

2016

AIP Conference Proceedings

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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 suggested 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, and has quadratic approximation order. In this paper we discuss an improvement of the triangular Shepard operator.

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On the bivariate Shepard–Lidstone operators

Cited by 28 publications

References 14 publications

A New Approximation Method with High Order Accuracy

A New Approximation Method with High Order Accuracy

Bivariate Shepard–Bernoulli operators

On the enhancement of the approximation order of triangular Shepard method

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