Curvelets, Wave Atoms, and Wave Equations

Abstract

We argue that two specific wave packet families---curvelets and wave atoms---provide powerful tools for representing linear systems of hyperbolic differential equations with smooth, time-independent coefficients. In both cases, we prove that the matrix representation of the Green's function is sparse in the sense that the matrix entries decay nearly exponentially fast (i.e., faster than any negative polynomial), and well organized in the sense that the very few nonnegligible entries occur near a few shifted diagonals, whose location is predicted by geometrical optics. This result holds only when the basis elements obey a precise parabolic balance between oscillations and support size, shared by curvelets and wave atoms but not wavelets, Gabor atoms, or any other such transform. A physical interpretation of this result is that curvelets may be viewed as coherent waveforms with enough frequency localization so that they behave like waves but at the same time, with enough spatial localization so that they simultaneously behave like particles. We also provide fast digital implementations of tight frames of curvelets and wave atoms in two dimensions. In both cases the complexity is O(N² log N) flops for N-by-N Cartesian arrays, for forward as well as inverse transforms. Finally, we present a geometric strategy based on wave atoms for the numerical solution of wave equations in smoothly varying, 2D time-independent periodic media. Our algorithm is based on sparsity of the matrix representation of Green's function, as above, and also exploits its low-rank block structure after separation of the spatial indices. As a result, it becomes realistic to accurately build the full matrix exponential using repeated squaring, up to some time which is much larger than the CFL timestep. Once available, the wave atom representation of the Green's function can be used to perform 'upscaled' timestepping. We show numerical examples and prove complexity results based on a priori estimates of sparsity and separation ranks. They beat the O(N^3) bottleneck on an N-by-N grid, for a wide range of physically relevant situations. In practice, the current wave atom solver can become competitive over a pseudospectral method in the regime when the wave equation should be solved several times with different initial conditions, as in reflection seismology.