Fundamental Solutions of PDEs as Radial Basis Functions in Multivariate Interpolation

2007 ◽  
Vol 7 (4) ◽  
pp. 321-340
Author(s):  
A. Masjukov
Keyword(s):  
Radial Basis Functions ◽  
Interpolation Method ◽  
A Priori ◽  
Fundamental Solutions ◽  
Basis Functions ◽  
Scale Parameters ◽  
Radial Basis ◽  
Undecidable Problem ◽  
The Right ◽  
Smooth Approximations

AbstractFor bivariate and trivariate interpolation we propose in this paper a set of integrable radial basis functions (RBFs). These RBFs are found as fundamental solutions of appropriate PDEs and they are optimal in a special sense. The condition number of the interpolation matrices as well as the order of convergence of the inter- polation are estimated. Moreover, the proposed RBFs provide smooth approximations and approximate fulfillment of the interpolation conditions. This property allows us to avoid the undecidable problem of choosing the right scale parameter for the RBFs. Instead we propose an iterative procedure in which a sequence of improving approx- imations is obtained by means of a decreasing sequence of scale parameters in an a priori given range. The paper provides a few clear examples of the advantage of the proposed interpolation method.

A new a priori net length estimation technique for integrated circuits using radial basis functions

2013 ◽  
Vol 39 (4) ◽  
pp. 1204-1218 ◽  
Author(s):  
Amin Farshidi ◽  
Logan Rakai ◽  
Bardia Samimi ◽  
Laleh Behjat ◽  
David Westwick
Keyword(s):  
Integrated Circuits ◽  
Radial Basis Functions ◽  
A Priori ◽  
Basis Functions ◽  
Estimation Technique ◽  
Length Estimation ◽  
Radial Basis

scholarly journals Three-Dimensional Swirl Flow Velocity-Field Reconstruction Using a Neural Network With Radial Basis Functions

2001 ◽  
Vol 123 (4) ◽  
pp. 920-927 ◽  
Author(s):  
J. Pruvost ◽  
J. Legrand ◽  
P. Legentilhomme
Keyword(s):  
Neural Network ◽  
Velocity Field ◽  
Radial Basis Functions ◽  
Swirling Flow ◽  
Interpolation Method ◽  
Measurement Techniques ◽  
Three Dimensional ◽  
Basis Functions ◽  
Hydrodynamic Characteristics ◽  
Radial Basis

For many studies, knowledge of continuous evolution of hydrodynamic characteristics is useful but generally measurement techniques provide only discrete information. In the case of complex flows, usual numerical interpolating methods appear to be not adapted, as for the free decaying swirling flow presented in this study. The three-dimensional motion involved induces a spatial dependent velocity-field. Thus, the interpolating method has to be three-dimensional and to take into account possible flow nonlinearity, making common methods unsuitable. A different interpolation method is thus proposed, based on a neural network algorithm with Radial Basis Functions.

scholarly journals An Approach Towards Generating Surrogate Models by Using RBFN With a Priori Bias

2014 ◽  
Author(s):  
Kaveh Amouzgar ◽  
Niclas Stromberg
Keyword(s):  
Radial Basis Functions ◽  
Large Scale ◽  
A Priori ◽  
Surrogate Models ◽  
Basis Functions ◽  
Normal Equation ◽  
A Posteriori ◽  
Mathematical Functions ◽  
Radial Basis ◽  
Price Functions

In this paper, an approach to generate surrogate models constructed by radial basis function networks (RBFN) with a priori bias is presented. RBFN as a weighted combination of radial basis functions only, might become singular and no interpolation is found. The standard approach to avoid this is to add a polynomial bias, where the bias is defined by imposing orthogonality conditions between the weights of the radial basis functions and the polynomial basis functions. Here, in the proposed a priori approach, the regression coefficients of the polynomial bias are simply calculated by using the normal equation without any need of the extra orthogonality prerequisite. In addition to the simplicity of this approach, the method has also proven to predict the actual functions more accurately compared to the RBFN with a posteriori bias. Several test functions, including Rosenbrock, Branin-Hoo, Goldstein-Price functions and two mathematical functions (one large scale), are used to evaluate the performance of the proposed method by conducting a comparison study and error analysis between the RBFN with a priori and a posteriori known biases. Furthermore, the aforementioned approaches are applied to an engineering design problem, that is modeling of the material properties of a three phase spherical graphite iron (SGI). The corresponding surrogate models are presented and compared.

scholarly journals On the Factors Affecting the Accuracy and Robustness of Smoothed-Radial Point Interpolation Method

2016 ◽  
Vol 9 (1) ◽  
pp. 43-72 ◽  
Author(s):  
Abderrachid Hamrani ◽  
Idir Belaidi ◽  
Eric Monteiro ◽  
Philippe Lorong
Keyword(s):  
Radial Basis Functions ◽  
Interpolation Method ◽  
Basis Functions ◽  
Integration Scheme ◽  
Shape Parameters ◽  
Radial Point Interpolation Method ◽  
Radial Point Interpolation ◽  
Radial Basis ◽  
Point Interpolation Method ◽  
Point Interpolation

AbstractIn order to overcome the possible singularity associated with the Point Interpolation Method (PIM), the Radial Point Interpolation Method (RPIM) was proposed by G. R. Liu. Radial basis functions (RBF) was used in RPIM as basis functions for interpolation. All these radial basis functions include shape parameters. The choice of these shape parameters has been and stays a problematic theme in RBF approximation and interpolation theory. The object of this study is to contribute to the analysis of how these shape parameters affect the accuracy of the radial PIM. The RPIM is studied based on the global Galerkin weak form performed using two integration technics: classical Gaussian integration and the strain smoothing integration scheme. The numerical performance of this method is tested on their behavior on curve fitting, and on three elastic mechanical problems with regular or irregular nodes distributions. A range of recommended shape parameters is obtained from the analysis of different error indexes and also the condition number of the matrix system. All resulting RPIM methods perform very well in term of numerical computation. The Smoothed Radial Point Interpolation Method (SRPIM) shows a higher accuracy, especially in a situation of distorted node scheme.

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