A family of multivariate multiquadric quasi-interpolation operators with higher degree polynomial reproduction

Wu, Ruiyi; Wu, Tieru; Li, Huilai

doi:10.1016/j.cam.2014.07.008

Cited by 10 publications

(4 citation statements)

References 15 publications

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“…Refs. [1,3,4,7,13,15] mainly researched the polynomial interpolation method of two variables, in which [13] deducted the polynomial interpolation with nodes in triangular arrangement and has a similar idea with this article. Refs.…”

Section: Introductionmentioning

confidence: 93%

Linear interpolation for spatial data grid by newton polynomials

Jiang¹,

Hu²,

Wu³

et al. 2016

J. Math. Computer Sci.

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This article concerns with an improved interpolation method based on Newton scheme with high dimensional data. It obtains the interpolation function when the nodes are arranged in a spatial grid. The spatial interpolation method is systematic and very useful in engineering field. As an example of application, we use the method to simulate the 3-D temperature field nearby a motor by 27 nodal temperature values. The results show that the interpolation function is quickly obtained within a limited cube space which realized data visualization. This demonstrates the effectiveness of the proposed method.

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Section: Introductionmentioning

confidence: 93%

Linear interpolation for spatial data grid by newton polynomials

Jiang¹,

Hu²,

Wu³

et al. 2016

J. Math. Computer Sci.

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“…Besides the high accuracy, the MQ quasi-interpolation has been proven to possess many other favorable properties, such as monotonicity preservation, shape preservation, and variation diminishing [17,18]. Because of these positive properties, researchers began to extend the univariate quasi-interpolation to multi-dimension [24][25][26].…”

Section: Bivariate Mq Quasi-interpolationsmentioning

confidence: 99%

“…MQ function was firstly introduced by Hardy [16], then it has been investigated extensively [17][18][19] and successfully applied to solve one dimensional (1-D) PDEs [12,[20][21][22][23]. Due to the extraordinary performance in 1-D, Ling [24] extended the univariate MQ quasi-interpolation to 2-D. Feng and Zhou [25], Wu et al [26] and Wu et al [27,28] constructed high-dimensional quasi-interpolations.…”

mentioning

confidence: 99%

Numerical Solution of High-Dimensional Shockwave Equations by Bivariate Multi-Quadric Quasi-Interpolation

2019

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Radial basis function-based quasi-interpolation performs efficiently in high-dimensional approximation and its applications, which can attain the approximant and its derivatives directly without solving any large-scale linear system. In this paper, the bivariate multi-quadrics (MQ) quasi-interpolation is used to simulate two-dimensional (2-D) Burgers' equation. Specifically, the spatial derivatives are approximated by using the quasi-interpolation, and the time derivatives are approximated by forward finite difference method. One advantage of the proposed scheme is its simplicity and easy implementation. More importantly, the proposed scheme opens the gate to meshless adaptive moving knots methods for the high-dimensional partial differential equations (PDEs) with shock or soliton waves. The scheme is also applicable to other non-linear high-dimensional PDEs. Two numerical examples of Burgers' equation (shock wave equation) and one example of the Sine-Gordon equation (soliton wave equation) are presented to verify the high accuracy and efficiency of this method.

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“…Ling proposed a multidimensional quasi-interpolation operator using the dimension-splitting multiquadric basis function approach [32], and Wu et al modified their idea by using multivariate divided difference and the idea of the superposition [33].…”

mentioning

confidence: 99%

A shape preserving quasi-interpolation operator based on a new transcendental RBF

Heidari¹,

Mohammadi²,

Marchi³

2021

Preprint

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It is well-known that the univariate Multiquadric quasi-interpolation operator is constructed based on the piecewise linear interpolation by |x|. In this paper, we first introduce a new transcendental RBF based on the hyperbolic tangent function as an smooth approximant to φ(r) = r with higher accuracy and better convergence properties than the MQ RBF √ r 2 + c 2 . Then the Wu-Schaback's quasi-interpolation formula is rewritten using the proposed RBF. It preserves convexity and monotonicity. We prove that the proposed scheme converges with a rate of O(h 2 ). So it has a higher degree of smoothness. Some numerical experiments are given in order to demonstrate the efficiency and accuracy of the method.

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A family of multivariate multiquadric quasi-interpolation operators with higher degree polynomial reproduction

Cited by 10 publications

References 15 publications

Linear interpolation for spatial data grid by newton polynomials

Linear interpolation for spatial data grid by newton polynomials

Numerical Solution of High-Dimensional Shockwave Equations by Bivariate Multi-Quadric Quasi-Interpolation

A shape preserving quasi-interpolation operator based on a new transcendental RBF

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