Radial basis functions

Acta Numerica ◽  
2000 ◽  
Vol 9 ◽  
pp. 1-38 ◽  
Author(s):  
M. D. Buhmann
Keyword(s):  
Radial Basis Function ◽  
Radial Basis Functions ◽  
Basis Function ◽  
Convergence Rates ◽  
Basis Functions ◽  
Point Of View ◽  
Theoretical Point ◽  
Grid Data ◽  
Radial Basis ◽  
Recent Developments

Radial basis function methods are modern ways to approximate multivariate functions, especially in the absence of grid data. They have been known, tested and analysed for several years now and many positive properties have been identified. This paper gives a selective but up-to-date survey of several recent developments that explains their usefulness from the theoretical point of view and contributes useful new classes of radial basis function. We consider particularly the new results on convergence rates of interpolation with radial basis functions, as well as some of the various achievements on approximation on spheres, and the efficient numerical computation of interpolants for very large sets of data. Several examples of useful applications are stated at the end of the paper.

Multilevel RBF collocation method for the fourth-order thin plate problem

2020 ◽  
pp. 2050079
Author(s):  
Lanling Ding ◽  
Zhiyong Liu ◽  
Qiuyan Xu
Keyword(s):  
Radial Basis Function ◽  
Collocation Method ◽  
Radial Basis Functions ◽  
Thin Plate ◽  
Fourth Order ◽  
Basis Function ◽  
Research Method ◽  
Meshfree Method ◽  
Basis Functions ◽  
Radial Basis

The radial basis functions meshfree method is a research method for thin plate problem which has gradually developed into a more mature meshfree method. It includes finite element, radial basis functions meshfree collocation method, etc. In this paper, we introduce the multilevel radial basis function collocation method for the fourth-order thin plate problem. We use nonsymmetric Kansa multilevel radial basis function collocation method to solve the fourth-order thin plate problem. Two numerical examples based on Wendland’s [Formula: see text] and [Formula: see text] functions are given to examine that the convergence of the multilevel radial basis function collocation method which is good for solving the fourth-order thin plate problem.

scholarly journals On the Numerical Approximation of Three-Dimensional Time Fractional Convection-Diffusion Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Kamran ◽  
Raheel Kamal ◽  
Gul Rahmat ◽  
Kamal Shah
Keyword(s):  
Laplace Transform ◽  
Radial Basis Function ◽  
Radial Basis Functions ◽  
Efficient Method ◽  
Basis Function ◽  
Numerical Scheme ◽  
Three Dimensional ◽  
Basis Functions ◽  
Convection Diffusion ◽  
Radial Basis

In this paper, we present an efficient method for the numerical investigation of three-dimensional non-integer-order convection-diffusion equation (CDE) based on radial basis functions (RBFs) in localized form and Laplace transform (LT). In our numerical scheme, first we transform the given problem into Laplace space using Laplace transform. Then, the local radial basis function (LRBF) method is employed to approximate the solution of the transformed problem. Finally, we represent the solution as an integral along a smooth curve in the complex left half plane. The integral is then evaluated to high accuracy by a quadrature rule. The Laplace transform is used to avoid the classical time marching procedure. The radial basis functions are important tools for scattered data interpolation and for solving partial differential equations (PDEs) of integer and non-integer order. The LRBF and global radial basis function (GRBF) are used to produce sparse collocation matrices which resolve the issue of the sensitivity of shape parameter and ill conditioning of system matrices and reduce the computational cost. The application of Laplace transformation often leads to the solution in complex plane which cannot be generally inverted. In this work, improved Talbot’s method is utilized which is an efficient method for the numerical inversion of Laplace transform. The stability and convergence of the method are discussed. Two test problems are considered to validate the numerical scheme. The numerical results highlight the efficiency and accuracy of the proposed method.

scholarly journals A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 65
Author(s):  
Benjamin Akers ◽  
Tony Liu ◽  
Jonah Reeger
Keyword(s):  
Radial Basis Function ◽  
Hilbert Transform ◽  
Basis Function ◽  
Differential Operators ◽  
Convergence Rates ◽  
Traveling Wave Solutions ◽  
Algebraic Decay ◽  
Pseudo Differential Operators ◽  
Radial Basis ◽  
The Hilbert Transform

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on R, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

scholarly journals Accurate radial basis functions technique for competitive and efficient solutions of non-linear Black-Scholes equations

2020 ◽  
Vol 20 (4) ◽  
pp. 60-83
Author(s):  
Vinícius Magalhães Pinto Marques ◽  
Gisele Tessari Santos ◽  
Mauri Fortes
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Efficient Solutions ◽  
Basis Functions ◽  
Adaptive Scheme ◽  
Call Options ◽  
Black Scholes ◽  
Non Linear ◽  
Radial Basis ◽  
Complex Models

ABSTRACTObjective: This article aims to solve the non-linear Black Scholes (BS) equation for European call options using Radial Basis Function (RBF) Multi-Quadratic (MQ) Method.Methodology / Approach: This work uses the MQ RBF method applied to the solution of two complex models of nonlinear BS equation for prices of European call options with modified volatility. Linear BS models are also solved to visualize the effects of modified volatility.  Additionally, an adaptive scheme is implemented in time based on the Runge-Kutta-Fehlberg (RKF) method.

scholarly journals The Conical Radial Basis Function for Partial Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
J. Zhang ◽  
F. Z. Wang ◽  
E. R. Hou
Keyword(s):  
Radial Basis Function ◽  
Boundary Value Problems ◽  
Basis Function ◽  
Boundary Value ◽  
Computational Cost ◽  
Basis Functions ◽  
Simple Scheme ◽  
Collocation Points ◽  
Radial Basis ◽  
Radial Basis Function Method

The performance of the parameter-free conical radial basis functions accompanied with the Chebyshev node generation is investigated for the solution of boundary value problems. In contrast to the traditional conical radial basis function method, where the collocation points are placed uniformly or quasi-uniformly in the physical domain of the boundary value problems in question, we consider three different Chebyshev-type schemes to generate the collocation points. This simple scheme improves accuracy of the method with no additional computational cost. Several numerical experiments are given to show the validity of the newly proposed method.

Differential Evolution-Based Optimization of Kernel Parameters in Radial Basis Function Networks for Classification

2013 ◽  
Vol 4 (1) ◽  
pp. 56-80 ◽  
Author(s):  
Ch. Sanjeev Kumar Dash ◽  
Ajit Kumar Behera ◽  
Satchidananda Dehuri ◽  
Sung-Bae Cho
Keyword(s):  
Neural Networks ◽  
Differential Evolution ◽  
Radial Basis Function ◽  
Basis Function ◽  
Learning Algorithm ◽  
Basis Functions ◽  
Second Phase ◽  
Rbf Neural Networks ◽  
Radial Basis ◽  
Two Phases

In this paper a two phases learning algorithm with a modified kernel for radial basis function neural networks is proposed for classification. In phase one a new meta-heuristic approach differential evolution is used to reveal the parameters of the modified kernel. The second phase focuses on optimization of weights for learning the networks. Further, a predefined set of basis functions is taken for empirical analysis of which basis function is better for which kind of domain. The simulation result shows that the proposed learning mechanism is evidently producing better classification accuracy vis-à-vis radial basis function neural networks (RBFNs) and genetic algorithm-radial basis function (GA-RBF) neural networks.

SIDE EFFECTS OF NORMALISING RADIAL BASIS FUNCTION NETWORKS

1996 ◽  
Vol 07 (02) ◽  
pp. 167-179 ◽  
Author(s):  
ROBERT SHORTEN ◽  
RODERICK MURRAY-SMITH
Keyword(s):  
Side Effects ◽  
Radial Basis Function ◽  
Basis Function ◽  
Function Approximation ◽  
Partition Of Unity ◽  
Basis Functions ◽  
Radial Basis Function Networks ◽  
Rbf Network ◽  
Linear Function Approximation ◽  
Radial Basis

Normalisation of the basis function activations in a Radial Basis Function (RBF) network is a common way of achieving the partition of unity often desired for modelling applications. It results in the basis functions covering the whole of the input space to the same degree. However, normalisation of the basis functions can lead to other effects which are sometimes less desirable for modelling applications. This paper describes some side effects of normalisation which fundamentally alter properties of the basis functions, e.g. the shape is no longer uniform, maxima of basis functions can be shifted from their centres, and the basis functions are no longer guaranteed to decrease monotonically as distance from their centre increases—in many cases basis functions can ‘reactivate’, i.e. re-appear far from the basis function centre. This paper examines how these phenomena occur, discusses their relevance for non-linear function approximation and examines the effect of normalisation on the network condition number and weights.

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