Hybrid Gaussian-cubic radial basis functions for scattered data interpolation.

The common approaches for scattered interpolations are use of polynomial and piece-wise polynomial spline, geostatistical methods, and radial basis functions, etc. The interpolation schemes using kriging and radial basis functions have the advantage of being meshless and dimensional independent. However, for the data sets having insufficient observations, radial basis functions have the advantage over geostatistical methods which requires variogram and statistical expertise. Moreover, radial basis functions can be used for scattered data interpolation with very good convergence, which is used for shape function interpolation in meshless methods for numerical solution of partial differential equations. For interpolation of large data sets, however, radial basis functions in their usual form, lead to solving an ill-conditioned system of equations for which a small error in the data can cause a significantly large error in the interpolated solution. In order to reduce this limitation of radial basis function interpolation schemes, we propose a hybrid kernel by using the conventional Gaussian and a shape parameter independent cubic radial basis function. Global particle swarm optimization method has been used to determine the optimal values of the shape parameter as well as the weight coefficients controlling the Gaussian and the cubic part in the hybridization. A series of numerical tests have been performed, which demonstrate that such hybridization stabilizes the interpolation scheme by yielding a far superior implementation compared to those obtained by using only the Gaussian or cubic radial basis function. The proposed kernel maintains the accuracy and stability at small shape parameter as well as relatively large degrees of freedom, which exhibit its potential for scattered data interpolation and application in science and engineering.