Papers › Multidimensional Phase Recovery and Interpolative Decomposition Butterfly Factorization
Multidimensional Phase Recovery and Interpolative Decomposition Butterfly Factorization
Ze Chen, Juan Zhang, Kenneth L. Ho, Haizhao Yang
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This paper focuses on the fast evaluation of the matvec g=Kf for K∈ℂ^(N×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)=e^(2πıΦ(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(NlogN) 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(NlogN) 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(NlogN) complexity once Φ is available. Numerical results are provided to demonstrate the effectiveness of the proposed framework.
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