Abstract

Geometrical (ray-based) techniques for high frequency wave scattering have been with us for centuries, but until relatively recently were applicable only to smooth scatterers. The Geometrical Theory of Diffraction (GTD), pioneered by Joe Keller and developed by numerous others since the 1960s, provided a powerful new methodology for scatterers with corners and sharp edges. It is a beautiful, wide-ranging and highly intuitive theory, inspired by physics but mathematically grounded in the theory of matched asymptotic expansions. GTD is an asymptotic theory. But it has also had significant influence on the direction of research into computational methods. Indeed, the past decade has seen exciting new developments in `hybrid numerical-asymptotic' (HNA) methods, which use FEM or BEM approximation spaces built from oscillatory basis functions, which are chosen by reference to the GTD . (In fact, Keller himself published a paper describing such a method.) For many basic scattering problems HNA methods achieve the `holy grail' of providing fixed accuracy with frequency-independent computational cost. In this talk I will outline the HNA approach and celebrate the ongoing role that Keller's GTD is playing in its development.