多维相位恢复与插值分解蝶形分解
Multidimensional Phase Recovery and Interpolative Decomposition Butterfly Factorization
摘要：
本文重点研究了$K\in \mathbb{C}^{N\times N}$的矩阵$g=Kf$的快速计算，$K\in \mathbb{C}^{N\times N}$是具有核函数$K(x,ξ)=e^{2\piıΦ(x,ξ)}$的多维振荡积分变换$g(x) = \int K(x,ξ) f(ξ)dξ$的离散化，其中$Φ(x,ξ)$是具有$x$和$ξ$的分段光滑相位函数，$\mathbb{R}^d$用于$d=2$或$3$。在只能间接访问相位函数$Φ$的情况下，引入了一个新的框架来计算具有$O(N\log N)$时间和存储复杂度的$Kf$。该框架由两个主要步骤组成：1）提出了一种从间接访问中恢复多维相位函数$Φ$的$O(N\log N)$算法; 2）设计了一种多维插值分解蝶形分解（MIDBF），一旦$Φ$可用，则以$O(N\log N)$的复杂度来评估矩阵$Kf$。数值结果证明了所提出的框架的有效性。
英文摘要：
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 $g(x) = \int K(x,ξ) f(ξ)dξ$ with a kernel function $K(x,ξ)=e^{2\piıΦ(x,ξ)}$, where $Φ(x,ξ)$ is a piecewise smooth phase function with $x$ and $ξ$ 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 $Φ$ is available. This framework consists of two main steps: 1) an $O(N\log N)$ algorithm for recovering the multidimensional phase function $Φ$ 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 $Φ$ is available. Numerical results are provided to demonstrate the effectiveness of the proposed framework.
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