# A Unified Framework for Oscillatory Integral Transform: When to use NUFFT or Butterfly Factorization?

## TL;DR

This paper introduces a unified framework for efficiently computing oscillatory integral transforms using NUFFT or butterfly factorization, with a method to choose the optimal approach and recover unknown functions.

## Contribution

It presents a novel $O(N ext{log}N)$ algorithm for recovering amplitude and phase functions from indirect data, and develops a stable low-rank butterfly factorization for fast evaluation.

## Key findings

- Framework achieves $O(N ext{log}N)$ complexity in evaluation.
- New low-rank matrix recovery algorithm for phase and amplitude functions.
- Numerical results confirm the effectiveness of the unified approach.

## Abstract

This paper concerns the fast evaluation of the matvec $g=Kf$ for $K\in \mathbb{C}^{N\times N}$, which is the discretization of the oscillatory integral transform $g(x) = \int K(x,\xi) f(\xi)d\xi$ with a kernel function $K(x,\xi)=\alpha(x,\xi)e^{2\pi\i \Phi(x,\xi)}$, where $\alpha(x,\xi)$ is a smooth amplitude function, and $\Phi(x,\xi)$ is a piecewise smooth phase function with $O(1)$ discontinuous points in $x$ and $\xi$. A unified framework is proposed to compute $Kf$ with $O(N\log N)$ time and memory complexity via the non-uniform fast Fourier transform (NUFFT) or the butterfly factorization (BF), together with an $O(N)$ fast algorithm to determine whether NUFFT or BF is more suitable. This framework works for two cases: 1) explicit formulas for the amplitude and phase functions are known, 2) only indirect access of the amplitude and phase functions are available. Especially in the case of indirect access, our main contributions are: 1) an $O(N\log N)$ algorithm for recovering the amplitude and phase functions is proposed based on a new low-rank matrix recovery algorithm, 2) a new stable and nearly optimal BF with amplitude and phase functions in a form of a low-rank factorization (IBF-MAT) is proposed to evaluate the matvec $Kf$. Numerical results are provided to demonstrate the effectiveness of the proposed framework.