Multi-scale discrete approximations of Fourier integral operators associated with canonical transformations and caustics

TL;DR

This paper presents a novel algorithm for efficiently computing Fourier integral operators related to canonical transformations, especially near caustics, using wave packets and local diffeomorphisms for improved accuracy.

Contribution

It introduces a universal operator representation and a wave packet-based computation method that handles singularities like caustics in Fourier integral operators.

Findings

Successfully applied to a wave equation parametrix near cusp singularity

Enables accurate discrete evaluation of Fourier integral operators with caustics

Provides a framework for handling singularities in canonical transformations

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 in the construction of a universal operator representation through the introduction of locally singularity-resolving diffeomorphisms, enabling the application of wave packet driven computation, and in the construction of the associated pseudo-differential 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.