发表机构
Fudan University（复旦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于蝶形分解和层级半可分矩阵的离散傅里叶积分算子近似求逆方法，具有近线性复杂度，可作为直接求解器或预处理器，数值实验验证了其有效性。
AI 中文摘要
本文提出了一种近似求逆离散傅里叶积分算子（FIOs）的新方法。给定一个FIO的$N \times N$矩阵表示$K$，所提出的算法包含两个阶段。在离线阶段，我们首先构造$K$的蝶形分解（BF）$\tilde{K}$，该分解支持快速的正向矩阵-向量乘法。然后，我们利用$\tilde{K}$和$\tilde{K}^{*}$对随机矩阵的快速应用，构造$G = K^{*} K$的层级半可分（HSS）近似$\tilde{G} \approx G$。最后，我们对HSS矩阵$\tilde{G}$应用ULV分解，得到近似$\tilde{F} \approx G^{-1}$。将这些近似组合起来，得到近似逆$K^{-1} \approx \tilde{F} \tilde{K}^{*}$。对于一维问题，离线阶段的复杂度为$O(N \log^{2} N)$；对于二维问题，复杂度为$O(N^{1.5} \log N)$。在线阶段，对于给定的输入向量$u$，所提出的方法以$O(N \log N)$的复杂度近似计算$K^{-1} u$，该复杂度适用于一维和二维问题。所提出的方法既可用作直接求解器，也可用作迭代方法的预处理器。一维和二维FIO的数值结果验证了所提出方法的有效性。
英文摘要
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 $\tilde{K}^{*}$ to random matrices. Finally, we apply the ULV factorization to the HSS matrix $\tilde{G}$ to obtain an approximation $\tilde{F} \approx G^{-1}$. Combining these approximations yields an approximate inverse $K^{-1} \approx \tilde{F} \tilde{K}^{*}$. 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.
Comments18 pages, 3 figures
