Multiscale Discrete Approximation of Fourier Integral Operators

Abstract. We develop a discretization and computational procedures for the approximation of the action of Fourier integral operators whose canonical relations are graphs. Such operators appear in many physical contexts and computational problems, for instance in the formulation of imaging and inverse scattering of seismic reflection data. Our discretization and algorithms are based on a multi-scale low-rank expansion of the action of Fourier integral operators using the dyadic parabolic decomposition of phase space, and on explicit constructions of low-rank separated representations that directly reflect the nature of such operators. The discretization and computational procedures explicitly connect and can be seamlessly overlaid to the discrete almost symmetric wave packet transformation. Numerical wave propagation and imaging examples illustrate our algorithms.