Multidimensional Phase Recovery and Interpolative Decomposition Butterfly Factorization

Abstract

This paper focuses on the fast evaluation of the matvec $g=Kf$ for $K\in {\mathbb{C}}^{N\times N}$, which is the discretization of a multidimensional oscillatory integral transform $g(x)=\int K(x,\xi )f(\xi )d\xi$ with a kernel function $K(x,\xi )=e^{2\pi ı\Phi (x,\xi )}$, where $\Phi (x,\xi )$ is a piecewise smooth phase function with $x$ and $\xi$ in ${\mathbb{R}}^d$ for $d=2$ or $3$. A new framework is introduced to compute $Kf$ with $O(N\log N)$ time and memory complexity in the case that only indirect access to the phase function $\Phi$ is available. This framework consists of two main steps: 1) an $O(N\log N)$ algorithm for recovering the multidimensional phase function $\Phi$ from indirect access is proposed; 2) a multidimensional interpolative decomposition butterfly factorization (MIDBF) is designed to evaluate the matvec $Kf$ with an $O(N\log N)$ complexity once $\Phi$ is available. Numerical results are provided to demonstrate the effectiveness of the proposed framework.