Erik Lehto: An introduction to radial basis function methods for solving differential equations

Approximation with radial basis functions (RBFs) originated from the multiquadric method introduced by Rolland Hardy in the late 1960s. Hardy was interested in reconstructing topography from survey data, and suggested using a linear combination of radially symmetric basis functions, each centered at a different data location, to interpolate the data. This collocation approach leads to a unique interpolant, regardless of the distribution of the data locations (in any number of dimensions), if the radial function satisfies certain requirements. A function of this kind is called an RBF, with examples such as the aforementioned multiquadric, the Gaussian and the thin plate spline.

Ed Kansa extended the method to solving partial differential equations in 1990, simply by replacing unknowns with RBF expansions and applying the (spatial) differential operators to the expansions. Kansa's method, or unsymmetric RBF collocation, has been shown to exhibit a number of desirable features, including spectral accuracy, algebraic simplicity and geometric flexibility. Despite these advantages, the method has not yet found wide-spread use (in the PDE context), possibly due to a high computational cost and some concerns regarding numerical stability.

In this presentation, I will give an introduction to RBF methods for solving differential equations, outlining key ideas and important theoretical results. The main focus will be on application examples of varying complexity, ranging from simple 1-D problems to the non-linear shallow water equations on the sphere. I will also highlight some ways of addressing the drawbacks mentioned above.