  This paper presents an efficient discontinuous Galerkin method to simulate
wave propagation in heterogeneous media with sub-cell variations. This method
is based on a weight-adjusted discontinuous Galerkin method (WADG), which
achieves high order accuracy for arbitrary heterogeneous media. However, the
computational cost of WADG grows rapidly with the order of approximation. In
this work, we propose a Bernstein-B\'ezier weight-adjusted discontinuous
Galerkin method (BBWADG) to address this cost. By approximating sub-cell
heterogeneities by a fixed degree polynomial, the main steps of WADG can be
expressed as polynomial multiplication and $L^2$ projection, which we carry out
using fast Bernstein algorithms. The proposed approach reduces the overall
computational complexity from $O(N^{2d})$ to $O(N^{d+1})$ in $d$ dimensions.
Numerical experiments illustrate the accuracy of the proposed approach, and
computational experiments for a GPU implementation of BBWADG verify that this
theoretical complexity is achieved in practice.
