Radial Basis Functions

The traditional basis functions in numerical PDEs are mostly coordinate functions, such as polynomial and trigonometric functions, which are computationally expensive in dealing with high dimensional problems due to their dependency on geometric complexity. Alternatively, radial basis functions (RBFs) are constructed in terms of one-dimensional distance variable irrespective of dimensionality of problems and appear to have a clear edge over the traditional basis functions directly in terms of coordinates. In the first part of this chapter, we introduces classical RBFs, such as globally-supported RBFs (Polyharmonic splines, Multiquadratics, Gaussian, etc.), and recently developed RBFs, such as compactly-supported RBFs. Following this, several problem-dependent RBFs, such as fundamental solutions, general solutions, harmonic functions, and particular solutions, are presented. Based on the second Green identity, we propose the kernel RBF-creating strategy to construct the appropriate RBFs.

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