Sparse Fourier Transform via Butterfly Algorithm
This work provides an efficient method for wave scattering and reflection seismology problems, but is incremental as it extends the butterfly algorithm with tensor-product acceleration.
The paper introduces a fast algorithm for computing sparse Fourier transforms on smooth curves or surfaces, achieving O(N log N) complexity with demonstrated accuracy in 2D and 3D numerical experiments.
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.