A unified framework for oscillatory integral transforms: When to use NUFFT or butterfly factorization?

Abstract

This paper concerns the fast evaluation of the matvec g=Kf for K∈CN×N, which is the discretization of an oscillatory integral transform g(x)=∫K(x,ξ)f(ξ)dξ with a kernel function K(x,ξ)=α(x,ξ)e2πiΦ(x,ξ), where α(x,ξ) is a smooth amplitude function, and Φ(x,ξ) is a piecewise smooth phase function with O(1) discontinuous points in x and ξ. A unified framework is proposed to compute Kf with O(Nlog⁡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(Nlog⁡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-MAT1) is proposed to evaluate the matvec Kf. Numerical results are provided to demonstrate the effectiveness of the proposed framework. 1IBF-MAT means interpolative butterfly factorization based on low-rank matrix information instead of explicit formulas used in the IBF in [22].