Shepard--Bernoulli operators

Caira, R.; Dell’Accio, Francesco

doi:10.1090/s0025-5718-06-01894-1

Cited by 25 publications

(18 citation statements)

References 22 publications

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“…From the numerical example, we can see that the accuracy is much better than Q d and power interpolation themselves. However, it would be interesting to compare the proposed method with other methods present in literature as a future work, as for example the operators proposed in [26,33], using appropriate test functions.…”

Section: Discussionmentioning

confidence: 99%

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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: Discussionmentioning

confidence: 99%

“…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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“…These functions were firstly proposed in [7] and result from adapting to the univariate case test functions generally used in the multivariate interpolation of large sets of scattered data [13]. We apply the approximating operators , see [7] for details.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Based on the idea in [7], we combine the multiquadric quasi-interpolation operator L B proposed in [3] with the generalized Taylor polynomial proposed in [8] to get a Bernoulli-type quasi-interpolation operator. The new operator could reproduce polynomials of higher degree than the operator L B .…”

Section: Introductionmentioning

confidence: 99%

A kind of Bernoulli-type quasi-interpolation operator with univariate multiquadrics

Wang¹,

Xu²

2010

Comput. Appl. Math.

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Abstract.In this paper, a kind of Bernoulli-type operator is proposed by combining a univariate multiquadric quasi-interpolation operator with the generalized Taylor polynomial. With an assumption on the shape-preserving parameter c, the convergence rate of the new operator is derived, which indicates that it could produce the desired precision. Numerical comparisons show that this method offers a higher degree of accuracy. Moreover, the associated algorithm is easily implemented.Mathematical subject classification: 41A05, 41A35.

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“…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].…”

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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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Shepard--Bernoulli operators

Cited by 25 publications

References 22 publications

A New Approximation Method with High Order Accuracy

A New Approximation Method with High Order Accuracy

A kind of Bernoulli-type quasi-interpolation operator with univariate multiquadrics

Bivariate Shepard–Bernoulli operators

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