Ray-Tracing the Ulam Way

Ray-tracing is a well established approach for modelling wave propagation at high frequencies, in which the ray trajectories are defined by a Hamiltonian system of ODEs. An approximation of the wave amplitude is then derived from estimating the density of rays in the neighbourhood of a given evaluation point. An alternative approach is to formulate the ray-tracing model directly in terms of the ray density in phase-space using the Liouville equation. The solutions may then be expressed in integral form using the Frobenius-Perron (F-P) operator, which is a transfer operator transporting the ray density along the trajectories. The classical approach for discretising such operators dates back to 1960 and the work of Stanislaw Ulam. The convergence of the Ulam method has been established in some cases, typically in low dimensional settings with continuous densities and hyperbolic dynamics. In this chapter, we outline some recent work investigating the convergence of the Ulam method for ray tracing in triangular billiards, where the dynamics are parabolic and the flow map contains jump discontinuities.