Radial Basis Function Interpolator

O.A. Estrada, I.D. Lopez-Gomez, C. Roldan, M. del P. Noriega, W.F. Florez, and T.A. Osswald. Numerical simulation of non-isothermal flow of non-newtonian incompressible fluids, considering viscous dissipation and inertia effects, using radial basis function interpolation. Numerical Methods for Heat and Fluid Flow, 2005. 

Omar Estrada, Ivan Lopez, Carlos Roldan, Maria del Pilar Noriega, and Whady Florez. Solution of steady and transient 2D-energy equation including convection and viscous dissipation effects using radial basis function interpolation. Journal of Applied Numerical Mathematics, 2005. 

The second approach is based on a model-free estimator the estimation is based on the adoption of a universal interpolator, i.e., a Radial Basis Function Interpolator (RBFI). Hence, differently from the previous approach, knowledge of the reaction kinetics is not required. 

When an online interpolator is used to estimate the uncertain term, the interpolation error g can be kept bounded, provided that a suitable interpolator structure is chosen [26, 28], Among universal approximators, Radial Basis Function Interpolators (RBFIs) provide good performance in the face of a relatively simple structure. Hence, Gaussian RBFs have been adopted, i.e.,... 

There is very little literature that accounts for discontinuities in radial basis function interpolation. However, a search radius method, in which a radial basis interpolant is used, rather than a pol3momial, would be practical... 

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. 

Radial basis functions. Radial interpolation uses radial basis functions in the linear combination that express the desired interpolated function, i.e.,... 

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,... 

Optimisation of radial basis function neural networks using biharmonic spline interpolation. Chemom. Intel Lab. Syst., 41, 17—29. 

The function ( ) is called a radial basis function if flic interpolation problem has a unique solution for any choice of data points. In some cases tiie polynomial term in Eq. (1) can be omitted and by combining it with Eq. (2), we obtain... 

Provided the inverse of ( ) exists, the solution w of the interpolation problem can be explicitly calculated, and has the form w = < )-1 y. The most popular and widely used radial basis function is the Gaussian basis function... 

Optimization of Radial Basis Function Neural Networks Using Bioharmonic Spline Interpolation. 

Tetteh and co-workers described the application of radial basis function (RBF) neural network models for property prediction and screening (114). They employed a network optimization strategy based on biharmonic spline interpolation for the selection of an optimum number of RBF neurons in the hidden layer and their associated spread parameter. Comparisons with the performance of a PLS regression model showed the superior predictive ability of the RBF neural model. 

The Method of Minimum Curvature Briggs, [27], introduced this method which has found widespread application in commercial mapping software (for example, see [84]). The method is known to be closely related to spline interpolation [90, Chapter 9]. As shown later, minimum curvature methods are a special case of the radial basis function approach and also of the kriging method. 

Radial Basis Functions In these methods a continuously differentiable, radial basis function, 4>, is introduced and interpolants of the form... 

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. 

A corollary of this result is that both the radial basis function method and the kriging method are maximum probability interpolants. 

The link between kriging, radial basis functions and maximum probability interpolants could be investigated in a much deeper way than in Section 4. The huge effort to analyse and develop radial basis function methods would be made more valuable if the participants in the growing radial basis function literature were more aware of the need for statistical considerations in the scattered data problem. Deterministic approaches to problems with sparse data are not applicable in most of the problems encountered in the geosciences. 

J. Carr, W.R. Fright, and R.K. Beatson (1997) Surface interpolation with radial basis functions for medical imaging. IEEE Transactions on Medical Imaging 16, 96-107. 

M. S. Eloater and A. iske (1996) Multistep scattered data interpolation using compactly supported radial basis functions. Journal of Computational and Applied Mathematics 73, 65-78. 

Interpolation consists of finding the correlation between the known points according to the selected basis functions. Hence, we need to search for appropriate equations that fit the behavior of our function / (x). For example, in linear interpolation, the chosen function is a straight line. The most commonly used functional forms are polynomials, rational functions, trigonometric functions and radial functions [10, 19, 21]. 

Once the basis functions are obtained, the points of the radial grid around each nucleus where the radial functions Rnt(r) are defined are interpolated, such that the values of the basis functions are obtained in the three-dimensional grid where the molecular or cluster calculation will be performed. 

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