Approximate Inversion of Discrete Fourier Integral Operators via Hierarchically Semiseparable Matrices

Abstract: This paper introduces a novel method for approximating the inverse of discrete Fourier integral operators (FIOs). Given an $N \times N$ matrix representation $K$ of an FIO, the proposed algorithm consists of two stages. In the offline stage, we first construct a butterfly factorization (BF) $\tilde{K}$ of $K$, which enables fast forward matrix-vector multiplication. We then construct a hierarchically semiseparable (HSS) approximation $\tilde{G} \approx G$, where $G = K^{*} K$, using fast applications of $\tilde{K}$ and $\tilde{K}<sup>{*}$ to random matrices. Finally, we apply the ULV factorization to the HSS matrix $\tilde{G}$ to obtain an approximation $\tilde{F} \approx G<sup>{-1}$. Combining these approximations yields an approximate inverse $$K^{-1} \approx \tilde{F} \tilde{K}<sup>{*}$$. The offline stage has complexity $O(N \log^{2} N)$ for 1D problems and $O(N^{1.5} \log N)$ for 2D problems. In the online stage, the proposed method approximates $K^{-1} u$ for a given input vector $u$ with complexity $O(N \log N)$ for both 1D and 2D problems. The proposed method can be used either as a direct solver or as a preconditioner for iterative methods. Numerical results for 1D and 2D FIOs demonstrate the effectiveness of the proposed method.