This paper focuses on the fast evaluation of the matrix-vector multiplication (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 $K(x,\xi )=e^{2\pi i\mathrm{\Phi }(x,\xi )}$, where $\mathrm{\Phi }(x,\xi )$ 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 $O(N\log \left(N\right))$ time and memories complexity in the case that only indirect access to the phase function Φ is available. This framework consists of two main steps: 1) an $O(N\log \left(N\right))$ 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 $O(N\log \left(N\right))$ complexity once Φ is available. Numerical results are provided to demonstrate the effectiveness of the proposed framework.

本文着重于矩阵向量乘法（matvec）的快速评估 $G=ķF$ 对于 $ķ\in {\mathbb{C}}^{ñ\times ñ}$，这是多维振荡积分变换的离散化 $G（X）=\int ķ（X，\xi ）F（\xi ）d\xi$ 具有内核功能 $ķ（X，\xi ）={Ë}^{2\pi \mathrm{一世}\mathrm{\Phi }（X，\xi ）}$，在哪里 $\mathrm{\Phi }（X，\xi ）$是具有分段平滑相位函数X和ξ在${\mathbb{[R}}^d$ 对于 $d=2$或3的新框架引入到计算Kf个与$Ø（ñ\mathrm{日志}（ñ））$在仅间接访问相位函数Φ的情况下，时间和存储器的复杂性。该框架包括两个主要步骤：1）$Ø（ñ\mathrm{日志}（ñ））$提出了一种从间接访问中恢复多维相位函数Φ的算法。2）多维内插分解蝴蝶因式分解（MIDBF）被设计来评估matvec Kf个用$Ø（ñ\mathrm{日志}（ñ））$Φ可用时的复杂度。数值结果表明了所提出框架的有效性。