Sparse Fourier Transform via Butterfly Algorithm

TL;DR

This paper presents a fast, accurate algorithm for computing sparse Fourier transforms on smooth surfaces, leveraging low-rank interactions and butterfly algorithms to achieve optimal complexity in wave scattering and seismology applications.

Contribution

The paper introduces a novel $O(N \,\log N)$ algorithm for sparse Fourier transforms on surfaces, combining butterfly algorithms with low-rank approximations and tensor-product properties.

Findings

Achieves $O(N \log N)$ computational complexity.

Demonstrates high accuracy in 2D and 3D numerical tests.

Effective in wave scattering and seismology problems.

Abstract

We introduce a fast algorithm for computing sparse Fourier transforms supported on smooth curves or surfaces. This problem appear naturally in several important problems in wave scattering and reflection seismology. The main observation is that the interaction between a frequency region and a spatial region is approximately low rank if the product of their radii are bounded by the maximum frequency. Based on this property, equivalent sources located at Cartesian grids are used to speed up the computation of the interaction between these two regions. The overall structure of our algorithm follows the recently-introduced butterfly algorithm. The computation is further accelerated by exploiting the tensor-product property of the Fourier kernel in two and three dimensions. The proposed algorithm is accurate and has an $O(N \log N)$ complexity. Finally, we present numerical results in both two and three dimensions.