Multivariate interpolation with increasingly flat radial basis functions of finite smoothness

In this paper, we consider multivariate interpolation with radial basis functions of finite smoothness. In particular, we show that interpolants by radial basis functions in ℝ ^d with finite smoothness of even order converge to a polyharmonic spline interpolant as the scale parameter of the radial basis functions goes to zero, i.e., the radial basis functions become increasingly flat.     

Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting

In this paper we discuss Sobolev bounds on functions that vanish at scattered points in a bounded, Lipschitz domain that satisfies a uniform interior cone condition. The Sobolev spaces involved may have fractional as well as integer order. We then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions (RBFs). These estimates include situations in which the target function does not belong to the native space of the RBF.

     

Limit problems for interpolation by analytic radial basis functions

Interpolation problems for analytic radial basis functions like the Gaussian and inverse multiquadrics can degenerate in two ways: the radial basis functions can be scaled to become increasingly flat, or the data points coalesce in the limit while the radial basis functions stay fixed. Both cases call for a careful regularization, which, if carried out explicitly, yields a preconditioning technique for the degenerating linear systems behind these interpolation problems. This paper deals with both cases. For the increasingly flat limit, we recover results by Larsson and Fornberg together with Lee, Yoon, and Yoon concerning convergence of interpolants towards polynomials. With slight modifications, the same technique can also handle scenarios with coalescing data points for fixed radial basis functions. The results show that the degenerating local Lagrange interpolation problems converge towards certain Hermite–Birkhoff problems. This is an important prerequisite for dealing with approximation by radial basis functions adaptively, using freely varying data sites.     

Error estimates of quasi-interpolation and its derivatives

Quasi-interpolation of radial basis functions on finite grids is a very useful strategy in approximation theory and its applications. A notable strongpoint of the strategy is to obtain directly the approximants without the need to solve any linear system of equations. For radial basis functions with Gaussian kernel, there have been more studies on the interpolation and quasi-interpolation on infinite grids. This paper investigates the approximation by quasi-interpolation operators with Gaussian kernel on the compact interval. The approximation errors for two classes of function with compact support sets are estimated. Furthermore, the approximation errors of derivatives of the approximants to the corresponding derivatives of the approximated functions are estimated. Finally, the numerical experiments are presented to confirm the accuracy of the approximations.     

Stability results for approximation by positive definite functions on SO(3)

We consider interpolation methods defined by positive definite functions on a locally compact group G. Estimates for the smallest and largest eigenvalue of the interpolation matrix in terms of the localization of the positive definite function on G are presented, and we provide a method to get positive definite functions explicitly on compact semisimple Lie groups. Finally, we apply our results to construct well-localized positive definite basis functions having nice stability properties on the rotation group SO(3).     

Spectral meshless radial point interpolation (SMRPI) method to two‐dimensional fractional telegraph equation

H. Ammari In this article, an innovative technique so‐called spectral meshless radial point interpolation (SMRPI) method is proposed and, as a test problem, is applied to a classical type of two‐dimensional time‐fractional telegraph equation defined by Caputo sense for (1 < α≤2). This new methods is based on meshless methods and benefits from spectral collocation ideas, but it does not belong to traditional meshless collocation methods. The point interpolation method with the help of radial basis functions is used to construct shape functions, which play as basis functions in the frame of SMRPI method. These basis functions have Kronecker delta function property. Evaluation of high‐order derivatives is not difficult by constructing operational matrices. In SMRPI method, it does not require any kind of integration locally or globally over small quadrature domains, which is essential of the finite element method (FEM) and those meshless methods based on Galerkin weak form. Also, it is not needed to determine strict value for the shape parameter, which plays an important role in collocation method based on the radial basis functions (Kansa's method). Therefore, computational costs of SMRPI method are less expensive. Two numerical examples are presented to show that SMRPI method has reliable rates of convergence. Copyright © 2015 John Wiley & Sons, Ltd.     

Multivariate Interpolation by Polynomials and Radial Basis Functions

In many cases, multivariate interpolation by smooth radial basis functions converges toward polynomial interpolants, when the basis functions are scaled to become flat. In particular, examples show and this paper proves that interpolation by scaled Gaussians converges toward the de Boor/Ron least polynomial interpolant. To arrive at this result, a few new tools are necessary. The link between radial basis functions and multivariate polynomials is provided by radial polynomials ||x-y||₂^2l\|x-y\|_2^{2\ell} that already occur in the seminal paper by C.A. Micchelli of 1986. We study the polynomial spaces spanned by linear combinations of shifts of radial polynomials and introduce the notion of a discrete moment basis to define a new well-posed multivariate polynomial interpolation process which is of minimal degree and also least and degree-reducing in the sense of de Boor and Ron. With these tools at hand, we generalize the de Boor/Ron interpolation process and show that it occurs as the limit of interpolation by Gaussian radial basis functions. As a byproduct, we get a stable method for preconditioning the matrices arising with interpolation by smooth radial basis functions.     

Error estimates and condition numbers for radial basis function interpolation

For interpolation of scattered multivariate data by radial basis functions, an “uncertainty relation” between the attainable error and the condition of the interpolation matrices is proven. It states that the error and the condition number cannot both be kept small. Bounds on the Lebesgue constants are obtained as a byproduct. A variation of the Narcowich-Ward theory of upper bounds on the norm of the inverse of the interpolation matrix is presented in order to handle the whole set of radial basis functions that are currently in use.     

Sobolev Error Estimates and a Bernstein Inequality for Scattered Data Interpolation via Radial Basis Functions

Error estimates for scattered-data interpolation via radial basis functions (RBFs) for target functions in the associated reproducing kernel Hilbert space (RKHS) have been known for a long time. Recently, these estimates have been extended to apply to certain classes of target functions generating the data which are outside the associated RKHS. However, these classes of functions still were not "large" enough to be applicable to a number of practical situations. In this paper we obtain Sobolev-type error estimates on compact regions of Rⁿ when the RBFs have Fourier transforms that decay algebraically. In addition, we derive a Bernstein inequality for spaces of finite shifts of an RBF in terms of the minimal separation parameter.     

Inverse and saturation theorems for radial basis function interpolation

While direct theorems for interpolation with radial basis functions are intensively investigated, little is known about inverse theorems so far. This paper deals with both inverse and saturation theorems. For an inverse theorem we especially show that a function that can be approximated sufficiently fast must belong to the native space of the basis function in use. In case of thin plate spline interpolation we also give certain saturation theorems.

     

Error estimates and condition numbers for radial basis function interpolation

For interpolation of scattered multivariate data by radial basis functions, an “uncertainty relation” between the attainable error and the condition of the interpolation matrices is proven. It states that the error and the condition number cannot both be kept small. Bounds on the Lebesgue constants are obtained as a byproduct. A variation of the Narcowich-Ward theory of upper bounds on the norm of the inverse of the interpolation matrix is presented in order to handle the whole set of radial basis functions that are currently in use.     

Análise de vigas laminadas utilizando o natural neighbour radial point interpolation method

In this work, a meshless method, “natural neighbour radial point interpolation method” (NNRPIM), is applied to the one‐dimensional analysis of laminated beams, considering the theory of Timoshenko.The NNRPIM combines the mathematical concept of natural neighbours with the radial point interpolation. Voronoï diagrams allows to impose the nodal connectivity and the construction of a background mesh for integration purposes, via influence cells. The construction of the NNRPIM interpolation functions is shown, and, for this, it is used the multiquadratic radial basis function. The generated interpolation functions possess infinite continuity and the delta Kronecker property, which facilitates the enforcement of boundary conditions, since these can be directly imposed, as in the finite element method (FEM).In order to obtain the displacements and the deformation fields, it is considered the Timoshenko theory for beams under transverse efforts. Several numerical examples of isotropic beams and laminated beams are presented in order to demonstrate the convergence and accuracy of the proposed application. The results obtained are compared with analytical solutions available in the literature.     

球面混合插值的逼近性质

球面径向基函数(SBF)和多项式样条函数均为处理球面散乱数据的有效工具. 本文考虑由球面径向基函数与球面多项式函数组成的混合插值模型, 并利用最小二乘法求解该模型. 对于该插值模型, 首先, 给出带Bessel势的Sobolev空间中的Bernstein不等式, 然后利用该不等式建立逼近正定理,并进一步给出该插值工具的误差估计. 最后, 研究该插值方式(即利用最小二乘法求解混合插值模型)的稳定性.     

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.     

A Radial Basis Function Method for Global Optimization

We introduce a method that aims to find the global minimum of a continuous nonconvex function on a compact subset of . It is assumed that function evaluations are expensive and that no additional information is available. Radial basis function interpolation is used to define a utility function. The maximizer of this function is the next point where the objective function is evaluated. We show that, for most types of radial basis functions that are considered in this paper, convergence can be achieved without further assumptions on the objective function. Besides, it turns out that our method is closely related to a statistical global optimization method, the P-algorithm. A general framework for both methods is presented. Finally, a few numerical examples show that on the set of Dixon-Szegö test functions our method yields favourable results in comparison to other global optimization methods.     

Thinning algorithms for scattered data interpolation

Multistep interpolation of scattered data by compactly supported radial basis functions requires hierarchical subsets of the data. This paper analyzes thinning algorithms for generating evenly distributed subsets of scattered data in a given domain in ℝ ^d .     

Numerical experiments on optimal shape parameters for radial basis functions

A numerical investigation on a technique for choosing an optimal shape parameter is proposed. Radial basis functions (RBFs) and their derivatives are used as interpolants in the asymmetric collocation radial basis method, for solving systems of partial differential equations. The shape parameter c in RBFs plays a major role in obtaining high quality solutions for boundary value problems. As c is a user defined value, inexperienced users may compromise the quality of the solution, often a problem of this meshless method. Here we propose a statistical technique to choose the shape parameter in radial basis functions. We use a cross‐validation technique suggested by Rippa 6 for interpolation problems to find a cost function Cost(c) that ideally has the same behavior as an error function. If that is the case, the parameter c that minimizes the cost function will be an optimal shape parameter, in the sense that it minimizes the error function. The form of the cost and error functions are analized for several examples, and for most cases the two functions have a similar behavior. The technique produced very accurate results, even with a small number of points and irregular grids. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Linear independence and divided derivatives of a Drinfeld module II

In this note we extend our previous results on the linear independence of values of the divided derivatives of exponential and quasi-periodic functions related to a Drinfeld module to divided derivatives of values of identity and quasi-periodic functions evaluated at the logarithm of an algebraic value. The change in point of view enables us to deal smoothly with divided derivatives of arbitrary order. Moreover we treat a full complement of quasi-periodic functions corresponding to a basis of de Rham cohomology.

     

Spherical Scattered Data Quasi-interpolation by Gaussian Radial Basis Function

Since the spherical Gaussian radial function is strictly positive definite, the authors use the linear combinations of translations of the Gaussian kernel to interpolate the scattered data on spheres in this article. Seeing that target functions are usually outside the native spaces, and that one has to solve a large scaled system of linear equations to obtain combinatorial coefficients of interpolant functions, the authors first probe into some problems about interpolation with Gaussian radial functions. Then they construct quasiinterpolation operators by Gaussian radial function, and get the degrees of approximation. Moreover, they show the error relations between quasi-interpolation and interpolation when they have the same basis functions. Finally, the authors discuss the construction and approximation of the quasi-interpolant with a local support function.     

函数的径向基表示

本文是关于函数的径向表示的综述性文章，基于我们对这方面的研究工作的了解，介绍了国际上近年来这方面的主要的有关研究结果以及在一些领域中应用的情况，文后附有主要的参考文献以便感兴趣的读者查阅。