A Fast Butterfly Algorithm for the Computation of Fourier Integral Operators

Abstract

This paper is concerned with the fast computation of Fourier integral operators of the general form R d e 2πıΦ(x,k) f (k)dk, where k is a frequency variable, Φ(x, k) is a phase function obeying a standard homogeneity condition, and f is a given input. This is of interest for such fundamental computations are connected with the problem of finding numerical solutions to wave equations, and also frequently arise in many applications including reflection seismology, curvilinear tomography and others. In two dimensions, when the input and output are sampled on N × N Cartesian grids, a direct evaluation requires O(N 4 ) operations, which is often times prohibitively expensive.This paper introduces a novel algorithm running in O(N 2 log N ) time, i. e. with near-optimal computational complexity, and whose overall structure follows that of the butterfly algorithm [30]. Underlying this algorithm is a mathematical insight concerning the restriction of the kernel e 2πıΦ(x,k) to subsets of the time and frequency domains. Whenever these subsets obey a simple geometric condition, the restricted kernel has approximately low-rank; we propose constructing such low-rank approximations using a special interpolation scheme, which prefactors the oscillatory component, interpolates the remaining nonoscillatory part and, lastly, remodulates the outcome. A byproduct of this scheme is that the whole algorithm is highly efficient in terms of memory requirement. Numerical results demonstrate the performance and illustrate the empirical properties of this algorithm.

Randomized estimation of spectral densities of large matrices made accurate

“…First, Theorem 4 suggests that for a strictly low rank matrix P , there is correspondence between the trace of the form in Eq. (18) and the solution of a generalized eigenvalue problem.…”

Computational aspects of the through-focus characteristics of the human eye

“…In the best known implementation this leads to the bi-dimensional discrete Fourier transform (we will refer to it as the FFT2-based approach), calculated via the Fast Fourier Transform algorithm. Its efficiency can be substantially enhanced using the fractional Fourier transform [29] or a "butterfly diagram" ideas [30], see also [31].…”

A fast butterfly algorithm for the hyperbolic Radon transform

“…Our algorithm for computing the summation in equation (4) follows that of Candès et al (2009). Readers are referred there for detailed mathematical exposition.…”