We present a new algorithm for solving the wave equation in smooth
inhomogeneous periodic media, which is unconstrained by the CFL condition
(which says that the timestep should be comparable to the spatial grid
spacing). We introduce an orthonormal basis of 'wave atoms', interpolating
between wavelets and Gabor, and whose key property is a precise balance
between oscillations and support called parabolic scaling. In that basis,
the time-dependent Green's function of the wave equation decomposes in a
sparse and separable way. As a result, it is possible to build the full
matrix exponential in optimal complexity up to some time which is much
bigger than the CFL timestep. Once available, this new representation can
be used to perform giant 'upscaled' time steps. We will show several 2D
numerical examples, as well as complexity results based on a priori
estimates of sparsity and separation ranks.

This is joint work with Emmanuel Candes and Lexing Ying.