An interval for the shape parameter in radial basis function
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An interval for the shape parameter in radial basis function
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8/8/2017
An interval for the shape parameter in radial basis function approximation
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Abstract Keywords 1. Introduction 2. Radial basis function approximation 3. A brief review of some strategies f… 4. An interval for the shape parameter 5. One and two dimensional interpol… 6. Selecting a good shape paramete… 7. Discussion and conclusion References
Applied Mathematics and Computation Volume 315, 15 December 2017, Pages 131–149
An interval for the shape parameter in radial basis function approximation Jafar Biazar
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, Mohammad Hosami
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https://doi.org/10.1016/j.amc.2017.07.047
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The radial basis functions (RBFs) depend on an auxiliary parameter, called the shape parameter. Great theoretical and numerical efforts have been made to find the relationship between the accuracy of the RBF-approximations and the value of the shape parameter. In many cases, the numerical approaches are based on minimization of an estimation of the error function, such as Rippa's approach and its modifications. These approaches determine a value for the shape parameter. In this paper, we propose a practical approach to determine an interval, instead of a value, without any minimization and estimation of error function. The idea is based on adding a loop on the shape parameter, but not to minimize an error norm. Suitable values of the shape parameter are determined by take into account the practical convergence behavior of the problem. The proposed method is applied to some illustrative examples, including one and twodimensional interpolation and partial differential equations. The results of the method are compared with those of some other approaches to confirm the reliability of the method. Furthermore, numerical stability of the method of line respect to the shape parameter is considered for time-dependent PDEs. The results show that, the proposed method can be used as an independent method to find a shape parameter, and also as an alternative for investigating the validity of the values of the other methods.
Keywords Radial basis function; Shape parameter; Kansa's method; Pseudo-spectral method; Method of line; Eigenvalue stability Choose an option to locate/access this article: Check if you have access through your login credentials or your institution
