CIMMS Lunchtime Series: Time Upscaling of Wave Equations Using Separable Atomic Decompositions

Laurent Demanet, Applied and Computational Mathematics, California Institute of Technology

We present a new algorithm for solving the wave equation in smooth inhomogeneous periodic media, which is unconstrained by the CFL condition. We introduce an orthonormal basis of multiscale 'wave atoms', whose key property is a precise balance between oscillations and support called the 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 low complexity up to some time which is much larger than the CFL timestep. Once available, this new representation can be used to perform giant 'upscaled' time steps. We expect applying this algorithm to reverse-time migration in the field of reflection seismology. We will show several 2D numerical examples, as well as complexity results based on a priori estimates of sparsity and separation ranks.