Approximation Using Gaussian Radial Basis Functions At Different Scales

Gaussian radial basis functions are a good archetype for high dimensional distributions. In this paper we will review radial basis function approximation, with some of its triumphs and pitfalls. We will then look at the use of Gaussians of different scales in the same approximation scheme in two different settings. One of them is the multilevel sparse grid setting, and the other is new work on emulation of the h-p finite element paradigm. In the former we will review recent work of the author and collaborators Manolis Georgoulis, Fuat Usta, Peter Dong, and Simon Hubbert. In the latter case we will discuss recent theoretical and numerical results developed by the lonely researcher (who is happy for anyone to join in).