Radial Basis Functions For Interpolation Research Articles

Order enhanced finite volume methods through non-polynomial approximation

In this paper, we introduce an approximation method that establishes certain order enhancements by leveraging radial basis functions (RBFs) in the numerical solution of conservation laws. The use of RBFs for interpolation and approximation is a well developed area of research. Of particular interest in this work is the development of high order finite volume (FV) weighted essentially non-oscillatory (WENO) methods, which utilize RBF approximations to obtain required data at cell interfaces. The aforementioned improvement in the order of accuracy is addressed through an analysis of the truncation error, resulting in expressions for the shape parameters appearing in the basis. This paper seeks to address the practical elements of the approach, including the evaluations of shape parameters as well as a hybrid implementation. To highlight the effectiveness of the non-polynomial basis in shock-capturing, the proposed methods are applied to systems of one-dimensional hyperbolic and weakly hyperbolic conservation laws and compared with several well-known WENO schemes in the literature. We also include a two-dimensional example for a scalar problem that demonstrates an extension to multiple dimensions. In the case of the non-smooth, weakly hyperbolic test problem, notable improvements are observed in predicting the location and height of the finite time blowup. The numerical results demonstrate that the proposed schemes attain notable improvements in accuracy, as indicated by the analysis of the reconstructions. A key contribution of this work is the development of robust third-order WENO method, which further demonstrates the effectiveness of the non-polynomial basis.

Modified local singular boundary method for solution of two-dimensional diffusion equation

This paper deals with a new version of the local singular boundary method (LSBM), which will allow the use of irregular node distribution and any shape of the solved area. Unlike previous modifications, it does not impose any requirements on the number of nodes in the support or the number of virtual points. Moreover, it uses radial basis functions for interpolation, not the weighted squares method. The article contains the derivation of the fundamental relations of the steady and unsteady state method and shows its effectiveness in four control examples.

An adaptive meshfree spectral graph wavelet method for partial differential equations

This paper proposes an adaptive meshfree spectral graph wavelet method to solve partial differential equations. The method uses radial basis functions for interpolation of functions and for approximation of the differential operators. It uses multiresolution analysis based on spectral graph wavelet for adaptivity. The set of scattered node points is subject to dynamic changes at run time which leads to adaptivity. The beauty of the method lies in the fact that the same operator is used for the approximation of differential operators and for the construction of spectral graph wavelet. Initially, we have applied the method on spherical diffusion equation. After that the problem of pattern formation on the surface of the sphere (using Turing equations) is addressed to test the strength of the method. The numerical results show that the method can accurately capture the emergence of the localized patterns at all the scales and the node arrangement is accordingly adapted. The convergence of the method is verified. For each test problem, the CPU time taken by the proposed method is compared with the CPU time taken by a traditional method (spectral method using radial basis functions). It is observed that the adaptive meshfree spectral graph wavelet method is highly efficient.

A Novel Method to Water Level Prediction using RBF and FFA

Water level prediction of rivers, especially in flood prone countries, can be helpful to reduce losses from flooding. A precise prediction method can issue a forewarning of the impending flood, to implement early evacuation measures, for residents near the river, when is required. To this end, we design a new method to predict water level of river. This approach relies on a novel method for prediction of water level named as RBF-FFA that is designed by utilizing firefly algorithm (FFA) to train the radial basis function (RBF) and (FFA) is used to interpolation RBF to predict the best solution. The predictions accuracy of the proposed RBF–FFA model is validated compared to those of support vector machine (SVM) and multilayer perceptron (MLP) models. In order to assess the models’ performance, we measured the coefficient of determination (R2), correlation coefficient (r), root mean square error (RMSE) and mean absolute percentage error (MAPE). The achieved results show that the developed RBF–FFA model provides more precise predictions compared to different ANNs, namely support vector machine (SVM) and multilayer perceptron (MLP). The performance of the proposed model is analyzed through simulated and real time water stage measurements. The results specify that the developed RBF–FFA model can be used as an efficient technique for accurate prediction of water stage of river.

An Effective Solution to Regression Problem by RBF Neuron Network

Radial Basis Function (RBF) neuron network is being applied widely in multivariate function regression. However, selection of neuron number for hidden layer and definition of suitable centre in order to produce a good regression network are still open problems which have been researched by many people. This article proposes to apply grid equally space nodes as the centre of hidden layer. Then, the authors use k-nearest neighbour method to define the value of regression function at the center and an interpolation RBF network training algorithm with equally spaced nodes to train the network. The experiments show the outstanding efficiency of regression function when the training data has Gauss white noise.

Efficient Algorithm for Training Interpolation RBF Networks With Equally Spaced Nodes

This brief paper proposes a new algorithm to train interpolation Gaussian radial basis function (RBF) networks in order to solve the problem of interpolating multivariate functions with equally spaced nodes. Based on an efficient two-phase algorithm recently proposed by the authors, Euclidean norm associated to Gaussian RBF is now replaced by a conveniently chosen Mahalanobis norm, that allows for directly computing the width parameters of Gaussian radial basis functions. The weighting parameters are then determined by a simple iterative method. The original two-phase algorithm becomes a one-phase one. Simulation results show that the generality of networks trained by this new algorithm is sensibly improved and the running time significantly reduced, especially when the number of nodes is large.

A novel efficient two-phase algorithm for training interpolation radial basis function networks

Interpolation radial basis function (RBF) networks have been widely used in various applications. The output layer weights are usually determined by minimizing the sum-of-squares error or by directly solving interpolation equations. When the number of interpolation nodes is large, these methods are time consuming, difficult to control the balance between the convergence rate and the generality, and difficult to reach a high accuracy. In this paper, we propose a two-phase algorithm for training interpolation RBF networks with bell-shaped basis functions. In the first phase, the width parameters of basis functions are determined by taking into account the tradeoff between the error and the convergence rate. Then, the output layer weights are determined by finding the fixed point of a given contraction transformation. The running time of this new algorithm is relatively short and the balance between the convergence rate and the generality is easily controlled by adjusting the involved parameters, while the error is made as small as desired. Also, its running time can be further enhanced thanks to the possibility to parallelize the proposed algorithm. Finally, its efficiency is illustrated by simulations.