Radial basis function interpolation

Radial basis function (RBF) interpolation is a method for constructing high-accuracy interpolants of unstructured data, possibly in high dimensions. The interpolant is a weighted sum of radial basis functions. RBF interpolation is a mesh-free method, meaning the nodes (points in the domain) need not lie on a structured grid or mesh, and does not require the formation of a mesh. It is often spectrally acurate and stable for large numbers of nodes even in high dimensions.

Many interpolation methods can be used as the theoretical foundation of algorithms for approximating linear operators, and RBF interpolation is no exception. RBF interpolation has been used to approximate differential operators, integral operators, and surface differential operators. These algorithms have been used to find highly accurate solutions of many differential equations including Navier-Stokes, Cahn-Hillard, and the shallow water equations.

Examples

The function $f(x)=\exp(x\cos(3\pi x))-1$ sampled at $n=15$ equally spaced nodes in the unit interval, and interpolated using the gaussian RBF with a shape parameter of $\varepsilon \approx 3.05048$.

The figure on the right depicts a radial basis function interpolation. The interpolant is visually the same everywhere except near the left boundary, where it is still very close.