Multidimensional phase recovery and interpolative decomposition butterfly factorization

•Optimally fast for amplitude and phase recovery of multidimensional oscillatory integral transforms with nonuniform grids.•Nearly optimally fast multidimensional interpolative decomposition butterfly factorization with nonuniform grids.•Applications in high-frequency integral equations. This paper focuses on the fast evaluation of the matrix-vector multiplication (matvec) g=Kf for K∈CN×N, which is the discretization of a multidimensional oscillatory integral transform g(x)=∫K(x,ξ)f(ξ)dξ with a kernel function K(x,ξ)=e2πiΦ(x,ξ), where Φ(x,ξ) is a piecewise smooth phase function with x and ξ in Rd for d=2 or 3. A new framework is introduced to compute Kf with O(Nlog⁡(N)) 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(Nlog⁡(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(Nlog⁡(N)) complexity once Φ is available. Numerical results are provided to demonstrate the effectiveness of the proposed framework.