几种基于散乱数据拟合的局部插值方法

本文首先针对散乱数据拟合的Shepard方法,结合截断多项式、B样条基函数和指数函数来构造其权函数,使新的权函数具有更高的光滑度和更好的衰减性,并且其光滑性和衰减性可以根据实际需要自由调节,从而提高了曲面的拟合质量．同时还给出一种类似的局部插值方法．另外,本文还基于多重二次插值,结合多元样条的思想,给出了两个局部插值算法．该算法较好地继承了多重二次插值曲面的性质,从而保证了拟合曲面具有好地光顺性和拟合精度．曲面整体也具有较高的光滑性．     

On the stationary Lp-approximation power to derivatives by radial basis function interpolation

In this paper, we consider approximation to derivatives of a function by using radial basis function interpolation. Most of well-known theories for this problem provide error analysis in terms of the so-called native space, say C_φ. However, if a basis function φ is smooth, the space C_φ is extremely small. Thus, the purpose of this study is to extend this result to functions in the homogenous Sobolev space.     

A meshless method for Burgers’ equation using MQ-RBF and high-order temporal approximation

In this paper, a new numerical method is proposed to solve one-dimensional Burgers’ equation using multiquadric (MQ) radial basis function (RBF) for spatial approximation and a second-order compact finite difference scheme for temporal approximation. The numerical results obtained by this way for different Reynolds number have been compared with the existing numerical schemes to show the accuracy and efficiency of the approach. To show the superiority of this meshless method, numerical experiments with non-uniform MQ interpolation node distribution are also performed.     

Performance evaluation of IDW,Kriging and multiquadric interpolation methods in producing noise mapping: A case study at the city of Isparta,Turkey

In order to produce an accurate noise map of a city or a region, it is necessary to make noise measurements at certain locations and these measurements must be modeled with the most suitable mathematical algorithm. A homogeneous and representative distribution of the noise measurement points is the first key factor in the production of sound noise maps. The second key element is the calculation of the noise values of gridding points based on noise measurement points according to the selected mathematical calculation method and the generation of maps according to these gridding points. In this study, a noise map of the Isparta city center and its periphery was produced using inverse distance weighted (IDW), Kriging and multiquadric interpolation methods with different parameters and four grid resolution. Then, the influence of parameter selection for each method was investigated in themself by taking into account grid resolution, namely 10 ∗ 10 m, 50 ∗ 50 m, 100 ∗ 100 m and 200 ∗ 200 m, and the performance of three method with 50 ∗ 50 m grid resolution were compared with each other. In addition, the noise mapping of the city of Isparta were produced by Kriging method with respect to maximum, average and minimum noise data and they were evaluated by considering the national environmental noise thresholds.     

A computational meshless method for the generalized Burger’s–Huxley equation

A numerical solution of the generalized Burger’s–Huxley equation, based on collocation method using Radial basis functions (RBFs), called Kansa’s approach is presented. The numerical results are compared with the exact solution, Adomian decomposition method (ADM) and Variational iteration method (VIM). Highly accurate and efficient results are obtained by RBFs method. Excellent agreement with the exact solution is observed while better (or same) accuracy is obtained than other numerical schemes cited in this work.     

Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled Burgers’ equations

This paper examines the numerical solution of the transient nonlinear coupled Burgers’ equations by a Local Radial Basis Functions Collocation Method (LRBFCM) for large values of Reynolds number (Re) up to 10³. The LRBFCM belongs to a class of truly meshless methods which do not need any underlying mesh but works on a set of uniform or random nodes without any a priori node to node connectivity. The time discretization is performed in an explicit way and collocation with the multiquadric radial basis functions (RBFs) are used to interpolate diffusion-convection variable and its spatial derivatives on decomposed domains. Five nodded domains of influence are used in the local support. Adaptive upwind technique [1] and [54] is used for stabilization of the method for large Re in the case of mixed boundary conditions. Accuracy of the method is assessed as a function of time and space discretizations. The method is tested on two benchmark problems having Dirichlet and mixed boundary conditions. The numerical solution obtained from the LRBFCM for different value of Re is compared with analytical solution as well as other numerical methods [15], [47] and [49]. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high Reynolds numbers.     

Optimal constant shape parameter for multiquadric based RBF-FD method

Radial basis functions (RBFs) have become a popular method for interpolation and solution of partial differential equations (PDEs). Many types of RBFs used in these problems contain a shape parameter, and there is much experimental evidence showing that accuracy strongly depends on the value of this shape parameter. In this paper, we focus on PDE problems solved with a multiquadric based RBF finite difference (RBF-FD) method. We propose an efficient algorithm to compute the optimal value of the shape parameter that minimizes the approximation error. The algorithm is based on analytical approximations to the local RBF-FD error derived in [1]. We show through several examples in 1D and 2D, both with structured and unstructured nodes, that very accurate solutions (compared to finite differences) can be achieved using the optimal value of the constant shape parameter.     

Unsteady seepage analysis using local radial basis function-based differential quadrature method

Numerical simulation of two-dimensional transient seepage is developed using radial basis function-based differential quadrature method (RBF-DQ). To the best of the authors’ knowledge, this is the first application of this method to seepage analysis. For the general case of irregular geometry and unstructured node distribution, the local form of RBF-DQ was used. The multiquadric type of radial basis functions was selected for the computations, and the results were compared with analytical, finite element method, and existing numerical solutions from the literature. Results of this study show that localized RBF-DQ can produce accurate results for the analysis of seepage. The method is meshfree and easy to program, but as with previous applications of RBFs, requires careful selection of suitable shape parameters. A practical method for estimating suitable shape parameters is discussed. For time integration, Crank–Nicolson, Galerkin and finite difference methods were applied, leading to stable results.     

A kind of multiquadric quasi-interpolation operator satisfying any degree polynomial reproduction property to scattered data

In this paper, by virtue of using the linear combinations of the shifts of f(x) to approximate the derivatives of f(x) and Waldron’s superposition idea (2009), we modify a multiquadric quasi-interpolation with the property of linear reproducing to scattered data on one-dimensional space, such that a kind of quasi-interpolation operator L_r+1f has the property of r+1(r∈Z,r≥0) degree polynomial reproducing and converges up to a rate of r+2. There is no demand for the derivatives of f in the proposed quasi-interpolation L_r+1f, so it does not increase the orders of smoothness of f. Finally, some numerical experiments are shown to compare the approximation capacity of our quasi-interpolation operators with that of Wu-Schaback’s quasi-interpolation scheme and Feng-Li’s quasi-interpolation scheme.