Compactly supported radial basis functions: How and why? - Zhu, S (SIAM, 2012)
The use of radial basis functions have attracted increasing attention in recent years as an elegant scheme for high-dimensional scattered data approximation, an practical method for machine learning, one of the foundations of mesh-free methods, an alternative way to construct higher order methods for solving partial dierential equations (PDEs), an emerging method for solving PDEs on surfaces, a novel method for mesh repair and so on. All these applications share one mathematical foundation: high dimensional approximation/interpolation. This paper explains why radial basis functions are preferred to multi-variate polynomials for scattered data approximation in high-dimensional space; and gives a brief description on how to construct the most commonly used compactly supported radial basis functions. Without sophisticated mathematics, one can construct a compactly supported (radial) basis function with required smoothness according to procedures described here. Short programs and tables for compactly supported radial basis functions are supplied.