Class Of Radial Basis Functions Research Articles

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We consider the interpolation problem for a class of radial basis functions (RBFs) that includes the classical polyharmonic splines (PHS). We show that the inverse of the system matrix for this interpolation problem can be approximated at an exponential rate in the block rank in the H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}$$\\end{document}-matrix format, if the block structure of the H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}$$\\end{document}-matrix arises from a standard clustering algorithm.

Complex Fourier element shape functions for analysis of 2D static and transient dynamic problems using dual reciprocity boundary element method

In this paper, the boundary element method is reformulated using new complex Fourier shape functions for solving two-dimensional (2D) elastostatic and dynamic problems. For approximating the geometry of boundaries and the state variables (displacements and tractions) of Navier's differential equation, the dual reciprocity (DR) boundary element method (BEM) is reconsidered by employing complex Fourier shape functions. After enriching a class of radial basis functions (RBFs), called complex Fourier RBFs, the interpolation functions of a complex Fourier boundary element framework are derived. To do so, polynomial terms are added to the functional expansion that only employs complex Fourier RBF in the approximation. In addition to polynomial function fields, the participation of exponential and trigonometric ones has also increased robustness and efficiency in the interpolation. Another interesting feature is that no Runge phenomenon happens in equispaced complex Fourier macroelements, unlike equispaced classic Lagrange ones. In the end, several numerical examples are solved to illustrate the efficiency and accuracy of the suggested complex Fourier shape functions and in comparison with the classic Lagrange ones, the proposed shape functions result in much more accurate and stable outcomes.

The solution of elastostatic and dynamic problems using the boundary element method based on spherical Hankel element framework

SummaryIn this paper, new spherical Hankel shape functions are used to reformulate boundary element method for 2‐dimensional elastostatic and elastodynamic problems. To this end, the dual reciprocity boundary element method is reconsidered by using new spherical Hankel shape functions to approximate the state variables (displacements and tractions) of Navier's differential equation. Using enrichment of a class of radial basis functions (RBFs), called spherical Hankel RBFs hereafter, the interpolation functions of a Hankel boundary element framework has been derived. For this purpose, polynomial terms are added to the functional expansion that only uses spherical Hankel RBF in the approximation. In addition to polynomial function fields, the participation of spherical Bessel function fields has also increased robustness and efficiency in the interpolation. It is very interesting that there is no Runge phenomenon in equispaced Hankel macroelements, unlike equispaced classic Lagrange ones. Several numerical examples are provided to demonstrate the effectiveness, robustness and accuracy of the proposed Hankel shape functions and in comparison with the classic Lagrange ones, they show much more accurate and stable results.

L p error estimates for approximation by Sobolev splines and Wendland functions on ℝ d

It is known that a Green's function-type condition may be used to derive rates for approximation by radial basis functions (RBFs). In this paper, we introduce a method for obtaining rates for approximation by functions which can be convolved with a finite Borel measure to form a Green's function. Following a description of the method, rates will be found for two classes of RBFs. Specifically, rates will be found for the Sobolev splines, which are Green's functions, and the perturbation technique will then be employed to determine rates for approximation by Wendland functions.

Meshless Pseudo-Polyharmonic Divergence-Free and Curl-Free Vector Fields Approximation

In this paper, we propose a meshless approximation of a vector field in multidimensional space minimizing quadratic forms related to the divergence or to the curl of a vector field. Our approach guarantees the conservation of the divergence-free or the curl-free properties, which are of great importance in applications. For instance, divergence-free vector fields correspond to incompressible fluid flows, and curl-free vector fields correspond to magnetic fields in the equations of classical electrodynamics. Our construction is based on the meshless approximation by the pseudo-polyharmonic functions which are a class of radial basis functions. We provide an Helmholtz–Hodge decomposition of the used native functional space. Numerical examples are included to illustrate our approach.

On Spherical Averages of Radial Basis Functions

A radial basis function (RBF) has the general form $$s(x)=\sum_{k=1}^{n}a_{k}\phi(x-b_{k}),\quad x\in\mathbb{R}^{d},$$ where the coefficients a 1,…,a n are real numbers, the points, or centres, b 1,…,b n lie in ℝ d , and φ:ℝ d →ℝ is a radially symmetric function. Such approximants are highly useful and enjoy rich theoretical properties; see, for instance (Buhmann, Radial Basis Functions: Theory and Implementations, [2003]; Fasshauer, Meshfree Approximation Methods with Matlab, [2007]; Light and Cheney, A Course in Approximation Theory, [2000]; or Wendland, Scattered Data Approximation, [2004]). The important special case of polyharmonic splines results when φ is the fundamental solution of the iterated Laplacian operator, and this class includes the Euclidean norm φ(x)=‖x‖ when d is an odd positive integer, the thin plate spline φ(x)=‖x‖2log ‖x‖ when d is an even positive integer, and univariate splines. Now B-splines generate a compactly supported basis for univariate spline spaces, but an analyticity argument implies that a nontrivial polyharmonic spline generated by (1.1) cannot be compactly supported when d>1. However, a pioneering paper of Jackson (Constr. Approx. 4:243–264, [1988]) established that the spherical average of a radial basis function generated by the Euclidean norm can be compactly supported when the centres and coefficients satisfy certain moment conditions; Jackson then used this compactly supported spherical average to construct approximate identities, with which he was then able to derive some of the earliest uniform convergence results for a class of radial basis functions. Our work extends this earlier analysis, but our technique is entirely novel, and applies to all polyharmonic splines. Furthermore, we observe that the technique provides yet another way to generate compactly supported, radially symmetric, positive definite functions. Specifically, we find that the spherical averaging operator commutes with the Fourier transform operator, and we are then able to identify Fourier transforms of compactly supported functions using the Paley–Wiener theorem. Furthermore, the use of Haar measure on compact Lie groups would not have occurred without frequent exposure to Iserles’s study of geometric integration.

The regularizing properties of anisotropic radial basis functions

In the present work we consider the problem of interpolating scattered data using radial basis functions (RBF). In general, it is well known that this leads to a discrete linear inverse problem that needs to be regularized in order to provide a meaningful solution. The work focuses on a metric-regularization approach, based on a new class of RBF, called anisotropic RBF. The work provides theoretical justifications for the regularization approach and it considers a suitable proposal for the metric, supporting it by numerical examples.

Dual Reciprocity Boundary Element Method for Magnetohydrodynamic Flow Using Radial Basis Functions

A dual reciprocity boundary element method is given to obtain the solution in terms of velocity and induced magnetic field for the study of MHD (magnetohydrodynamic) flow through a rectangular duct having insulating walls. The equations are transformed to two types of nonlinear Poisson equations and the right-hand sides in these equations are approximated using combinations of two classes of radial basis functions (the value of the function and its normal derivatives are utilized for approximation). Computations are carried out for several values of the Hartman number (0 h M h 10) by using constant boundary elements. Comparisons are made for two types of formulations and for traditional and osculatory type approximations of the right-hand side functions. It is found that osculatory interpolation gives better results than traditional interpolation and the type of the Poisson equation, which contains derivative of the unknown function, is better than the other type, which contains unknown function only. The results for velocity and induced magnetic field are illustrated by some selected graphs.

A quasi-interpolation method for solving stiff ordinary differential equations

Based on the idea of quasi-interpolation and radial basis functions approximation, a numerical method is developed to quasi-interpolate the forcing term of differential equations by using radial basis functions. A highly accurate approximation for the solution can then be obtained by solving the corresponding fundamental equation and a small size system of equations related to the initial or boundary conditions. This overcomes the ill-conditioning problem resulting from using the radial basis functions as a global interpolant. Error estimation is given for a particular second-order stiff differential equation with boundary layer. The result of computations indicates that the method can be applied to solve very stiff problems. With the use of multiquadric, a special class of radial basis functions, it has been shown that a reasonable choice for the optimal shape parameter is obtained by taking the same value of the shape parameter as the perturbed parameter contained in the stiff equation. Copyright © 2000 John Wiley & Sons, Ltd.

A new class of radial basis functions with compact support

Radial basis functions are well-known and successful tools for the interpolation of data in many dimensions. Several radial basis functions of compact support that give rise to nonsingular interpolation problems have been proposed, and in this paper we study a new, larger class of smooth radial functions of compact support which contains other compactly supported ones that were proposed earlier in the literature.

Multidimensional interpolation using osculatory radial basis functions

A procedure has been developed for the interpolation of functions defined in two dimensions using the values of the function and its normal derivatives at the boundary. The interpolating functions used are combinations of two classes of radial basis functions. This permits an interpolation scheme with osculatory features, i.e., the interpolant has a specified slope at the boundary interpolation points. The improved accuracy of the method is demonstrated over interpolation schemes using the traditional radial basis functions (with no osculation), especially for the calculation of the derivatives of the function at various internal points. A brief discussion on the utility of the new interpolation technique in solving nonlinear Poisson problems using the dual reciprocity boundary element method is provided.

Improved multiquadric approximation for partial differential equations

Based on the idea of the DRM, a numerical method has been devised to interpolate the forcing term of partial differential equations by using multiquadric approximations, a special class of radial basis functions, and then use them to approximate particular solutions. To obtain a good shape parameter of the multiquadrics, we use the technique of cross validation. After we find a particular solution, we then use the method of fundamental solutions to solve the homogeneous PDEs. To demonstrate the effectiveness of our method, four numerical results, including a 3D case, are given.

Radial basis function networks with FIR/IIR synapses

A learning algorithm for a special class of Radial Basis Functions (RBF) networks is proposed. The novelty aspect of the architecture consists in replacing the usual scalar values of the output weights with linear FIR or IIR filters transfer functions. Simulation results for approximating a lowpass Butterworth filter are presented and other potential temporal processing applications are outlined.

Scattered Hermite interpolation using radial basis functions

We study the scattered Hermite interpolation problem and find several classes of radial basis functions, including the multiquadrics, which may be implemented for this interpolation scheme.