A new generalization of multiquadric functions φ(x)=c2d+||x||2d, where x∈Rn, c∈R, d∈N, is presented to increase the accuracy of quasi-interpolation further. With the restriction to Euclidean spaces ...of odd dimensionality, the generalization can be used to generate a quasi-Lagrange operator that reproduces all polynomials of degree 2d−1. In contrast to the classical multiquadric, the convergence rate of the quasi-interpolation operator can be significantly improved by a factor h2d−n−1, where h>0 represents the grid spacing. Among other things, we compute the generalized Fourier transform of this new multiquadric function. Finally, an infinite regular grid is employed to analyse the properties of the aforementioned generalization in detail. We also present numerical results to demonstrate the advantages of our new multiquadric functions.
We propose a simple approach which considerably improves the performance of the well-known Kansa method for the solution of boundary value problems (BVPs). In the proposed approach, in contrast to ...the traditional Kansa method where the centres are placed in the closure of the domain of the BVP in question, the centres can be located outside it. We also employ a novel hybrid technique for the determination of the shape parameter in the radial basis functions used. Numerical examples in 2D and 3D are presented to demonstrate the effectiveness of the proposed method.
Based on the Uncertainty Principle of radial basis functions (RBFs), it is known that the condition number and the error cannot be both kept small at the same time. In contrast to the traditional ...condition number, the effective condition number provides a much better estimation of the actual condition number of the resultant matrix system. In this paper, motivated by the Uncertainty Principle of RBFs, we propose to apply the effective condition number as a numerical tool to determine a reasonably good shape parameter value in the context of the Kansa method coupled with the fictitious point method. Six examples for second and fourth order partial differential equations in 2D and 3D are presented to demonstrate the effectiveness of the proposed method.
We propose a simple approach to improve the accuracy of the Radial Basis Function Differential Quadrature (RBF-DQ) method for the solution of elliptic boundary value problems. While the traditional ...RBF-DQ method places the centers exclusively inside the domain, the proposed method expands the region for the centers allowing them to lie both inside and outside the computational domain. Furthermore, we seek an improvement to determine the shape parameter for the radial basis function by using the modified Franke’s formula to find an initial search interval for the leave-one-out cross-validation method, which is a widely used method for the determination of the shape parameter. Both 2D and 3D numerical examples are presented to demonstrate the effectiveness of the proposed method.
Continuously differentiable radial basis functions (C∞-RBFs) are the best method to solve numerically higher dimensional partial differential equations (PDEs). Among the reasons are:1.An ...n-dimensional problem becomes a one-dimensional radial distance problem,2.The convergence rate increases with the dimensionality,3.Such RBFs possess spectral convergence.Finitely supported polynomial methods only converge at polynomial rates. C∞-RBFs have global support; the systems of equations may become computationally singular if the condition number exceeds the inverse machine epsilon, εM. The solution to computational singularity is to decrease the effective εM by either hardware or software methods. Computer scientists developed rapidly executable multi-precision packages.
In this paper, we propose a new Kansa method with fictitious centre approach in which the radial basis function (RBF) approximation is augmented by polynomial basis functions. The proposed approach ...considerably improves not only the accuracy but also the stability of the recently proposed ghost point method using radial basis functions. The difficulty of selecting a good RBF shape parameter is no longer an issue. Two numerical examples are presented to show the effectiveness and the improvement of the proposed method over previous methods.
This article studies sufficient conditions on families of approximating kernels which provide N-term approximation errors from an associated nonlinear approximation space which match the best known ...orders of N-term wavelet expansion. These conditions provide a framework which encompasses some notable approximation kernels including splines, cardinal functions, and many radial basis functions such as the Gaussians and general multiquadrics. Examples of such kernels are given to justify the criteria. Additionally, the techniques involved allow for some new results on N-term Greedy interpolation of Sobolev functions via radial basis functions.
Abstract-Spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries. Since radial basis function (RBF) ...approxima-ions permit this even for multivariate scattered data, there has been much recent interest in practical algorithms to compute these approximations effectively.
Several types of RBFs feature a free parameter (e.g.,
c in the multiquadric (MQ) case
φ(
r) = √
r
2 +
c
2). The limit of
c → ∞ (increasingly flat basis functions) has not received much attention because it leads to a severely ill-conditioned problem. We present here an algorithm which avoids this difficulty, and which allows numerically stable computations of MQ RBF interpolants for all parameter values. We then find that the accuracy of the resulting approximations, in some cases, becomes orders of magnitude higher than was the case within the previously available parameter range.
Our new method provides the first tool for the numerical exploration of MQ RBF interpolants in the limit of
c → ∞. The method is in no way specific to MQ basis functions and can—without any change—be applied to many other cases as well.
In this study we apply a Kansa-radial basis function (RBF) collocation method to 2D and 3D boundary value problems (BVPs) governed by high order partial differential equations (PDEs) of order 2N ...where N∈N,N≥3. As in such problems there are N boundary conditions (BCs), N distinct sets of boundary centres are needed. These could all be placed on the boundary with each set being different to the other or, alternatively, each set of boundary centres could be placed on a corresponding distinct curve surrounding the boundary of the problem. We apply these approaches to several 2D and 3D high order BVPs.
We propose a Kansa-radial basis function (RBF) collocation method for the solution of 2D and 3D high order (i.e. of order 2N where N∈N,N≥3) boundary value problems (BVPs) in multiply connected ...domains. These problems include N boundary conditions (BCs), and thus we need to select N distinct sets of boundary centres. One set is positioned on the physical boundary of the problem while the rest are placed outside the exterior boundary of the domain. The efficacy of the proposed approach is demonstrated by applying it to a 2D and a 3D high order BVP in multiply connected domains.
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