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Scattered data interpolation schemes using kriging and radial basis functions(RBFs) have the advantage of being meshless and dimensional independent,however, for the data sets having insufficient observations, RBFs have theadvantage over geostatistical methods as the latter requires variogram studyand statistical expertise. Moreover, RBFs can be used for scattered datainterpolation with very good convergence, which makes them desirable for shapefunction interpolation in meshless methods for numerical solution of partialdifferential equations. For interpolation of large data sets, however, RBFs intheir usual form, lead to solving an ill-conditioned system of equations, forwhich, a small error in the data can cause a significantly large error in theinterpolated solution. In order to reduce this limitation, we propose a hybridkernel by using the conventional Gaussian and a shape parameter independentcubic kernel. Global particle swarm optimization method has been used toanalyze the optimal values of the shape parameter as well as the weightcoefficients controlling the Gaussian and the cubic part in the hybridization.Through a series of numerical tests, we demonstrate that such hybridizationstabilizes the interpolation scheme by yielding a far superior implementationcompared to those obtained by using only the Gaussian or cubic kernels. Theproposed kernel maintains the accuracy and stability at small shape parameteras well as relatively large degrees of freedom, which exhibit its potential forscattered data interpolation and intrigues its application in global as well aslocal meshless methods for numerical solution of PDEs.

Hybrid Gaussian-cubic radial basis functions for scattered data interpolation