Radial basis function technique

B. Waiczack and D.L. Massart, The radial basis functions — partial least squares approach as a flexible non-linear regression technique. Anal. Chim. Acta, 331 (1996) 177-185. 

The aim of this part of the book is to present the main and current numerical techniques that are used in polymer processesing. This chapter presents basic principles, such as error, interpolation and numerical integration, that serve as a foundation to numerical techniques, such as finite differences, finite elements, boundary elements, and radial basis functions collocation methods. 

In Chapter 11 of this book we will use the thin spline radial function to develop the radial basis functions collocation method (RBFCM). A well known property of radial interpolation is that it renders a convenient way to calculate derivatives of the interpolated function. This is an advantage over other interpolation functions and it is used in other methods such us the dual reciprocity boundary elements [43], collocation techniques [24], RBFCM, etc. For an interpolated function u,... 

In this section, we implement the radial basis function method in the energy equation and apply the technique to an example problem. We begin with a steady-state energy balance given by... 

The second part of training radial basis function networks assumes that the number of basis functions, i.e., the number of hidden units, and their center and variability parameters have been determined. Then all that remains is to find the linear combination of weights that produce the desired output (target) values for each input vector. Since this is a linear problem, convergence is guaranteed and computation proceeds rapidly. This task can be accomplished with an iterative technique based on the perception training rule, or with various other numerical techniques. Technically, the problem is a matrix inversion problem ... 

Multiple concentration fields are used here in an attempt to capture the dominant effects brought about by equations 1 and 2 on the spatially discrete anodic and cathodic areas formed during exposure to 5% NaCl solution. Multiion electrolyte simulation has also been documented [12] using a nested radial basis function (RBF) approach to predict concentration profiles around a rotating disk. Here, the evolution of Mr", [02], [OH ] and [H+] fields around a planar interface is predicted, governed by corrosion rates determined using data obtained from the rotating disc technique. 

A way to overcome this problem is to generate an approximation of complex analysis code that describes the process accurately, but at a much lower cost. Metamodels offer an approximation in that they provide a model of the model . Clarke et al. (2005) [58] suggested metamodelhng techniques, namely response surface methodology (RSM), radial basis function (RBF), kriging model and multivariate adaptive regression sphnes (MARS) as potentially useful approaches. Computer deterministic experiments have been addressed by Charles et al. (1996) [59], Simpson et al. (1998) [60], CappeUeri et al. (2002) [61] and Aguire et al. (2(X)7) [62],... 

Aoyama and Ichikawa have given analytic formulas for the partial derivatives of the output value of either a HL or output layer PE with respect to the input value of an input PE. Their formulas, which are applicable to any feedforward network with differentiable transfer functions in the hidden and output layers, allow you to give a precise, analytical answer to the question that sensitivity analysis asks (see above). A similar sensitivity analysis has been performed for a radial basis function ANN. Aoyama and coworkers introduced another technique useful in network analysis the reconstruction of weight matrices for a backpropagation network. They used a learning... 

Selection of the supervised classification technique or the combination of techniques suitable for accomplishing the classification task. Popular supervised classifiers are Multi-Layer Perceptron Artificial Neural Networks (MLP-ANN), Support Vector Machines (SVM), k-Nearest Neighbours (k-NN), combinations of genetic algorithms (GA) for feature selection with Linear Discriminant Analysis (LDA), Decision Trees and Radial Basis Function (RBF) classifiers. 

A new method for permeability prediction of carbonate formations with their very complex pore structures was published by Trevizan et al. (2014). From core permeability data and NMR measurements, a radial-basis-function (RBF) prediction model was developed. The model analyses a small number of principal components instead of the full spectrum. This robust technique was applied on offshore Brazil wells. 

Thus for large sets of scattered data and with a need to evaluate the interpolant at a large number of points on a grid one might be better served, when D is small, by the numerical solution of a partial differential equation as in the Briggs technique. However, the recent research into radial basis functions with compact support [61] and application of the fast multipole method [29] do provide efficient methods. 

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