  We present a construction of a multi-scale Gaussian beam parametrix for the
Dirichlet boundary value problem associated with the wave equation, and study
its convergence rate to the true solution in the highly oscillatory regime. The
construction elaborates on the wave-atom parametrix of Bao, Qian, Ying, and
Zhang and extends to a multi-scale setting the technique of Gaussian beam
propagation from a boundary of Katchalov, Kurylev and Lassas.
