Multiscale discrete approximations of Fourier integral operators associated with canonical transformations and caustics

Abstract

We develop an algorithm for the computation of general Fourier integral operators associated with canonical graphs. The algorithm is based on dyadic parabolic decomposition using wave packets and enables the discrete approximate evaluation of the action of such operators on data in the presence of caustics. The procedure consists of constructing a universal operator representation through the introduction of locally singularity-resolving diffeomorphisms, thus enabling the application of wave packet--driven computation, and of constructing the associated pseudodifferential joint-partition of unity on the canonical graphs. We apply the method to a parametrix of the wave equation in the vicinity of a cusp singularity.