On the selection of a better radial basis function and its shape parameter in interpolation problems - Chen, Chuin-Shan; Noorizadegan, Amir; Young, D.L.; Chen, C.S. (Applied mathematics and computation, 2023)
•A highly accurate result for the shape parameter corresponds to the maximum effective condition number.•The effective condition number helps to predict the error behavior regarding the type of RBF.•The effective condition number leads to an error estimation with respect to the function to be interpolated.•Ten test functions are interpolated using the multiquadric, Matern family, and Gaussian basis functions to show the advantage of the proposed method (ECONM) compared with the leave-one-out cross-validation (LOOCV) algorithm. A traditional criterion to calculate the numerical stability of the interpolation matrix is its standard condition number. In this paper, it is observed that the effective condition number (κeff) is more informative than the standard condition number (κ) in investigating the numerical stability of the interpolation problem. While the κeff considers the function to be interpolated, the standard condition number only depends on the interpolation matrix. We propose using the shape parameter corresponding to the maximum κeff to obtain a small error in RBF interpolation. It is also observed that the κeff helps to predict the error behavior with respect to the type of the RBF, where the basis function with a higher effective condition number yields a smaller error. In the end, we conclude that the effective condition number links to the error with respect to the selection of a radial basis function, choosing its shape parameter, number of collocation points, and test function. To this end, ten test functions are interpolated using the multiquadric, Matern family, and Gaussian basis functions to show the advantage of the proposed method.