用于快速高阶计算一般域上积分与卷积的截断傅里叶滤波（Truncated Fourier Filtering, TFF）方法
"Truncated Fourier Filtering" method for fast and high-order evaluation of integrals and convolutions in general domains
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中文总结 AI 辅助
本文提出截断傅里叶滤波（TFF）算法，可快速高阶计算一般m维域上的积分与卷积，复杂度低且对复杂几何结构有超代数收敛性。
中文摘要 AI 辅助
本文介绍并分析了一种新型算法——截断傅里叶滤波（Truncated Fourier Filtering, TFF），该算法基于m维笛卡尔网格，可快速、高阶精度地计算一般m维域（m≥1）上涉及分段光滑（可能不连续）被积函数的标准积分与卷积。对于N点离散化，该方法在单次O(N log N)预计算步骤（m=1维时无需该步骤）后，标准积分的计算复杂度为O(N)，卷积的计算复杂度为O(N log N)。TFF的核心思想是，通过在适当扩展的周期域上对积分域的特征函数进行截断傅里叶展开来近似该特征函数，并通过笛卡尔网格上具有适当选择离散化尺寸的梯形求积法计算所得积分。尽管概念简单，TFF即使对于复杂、可能非光滑甚至非利普希茨（non-Lipschitz）的几何结构也能达到高阶精度。本文提供了完整的理论分析，证明了该整体方法具有超代数收敛性（即收敛速度快于N的任何负次幂）。
英文摘要
This paper introduces and analyzes a novel algorithm---Truncated Fourier Filtering (TFF)---for the fast, high-order accurate evaluation of standard integrals and convolutions involving piecewise-smooth (possibly discontinuous) integrands over general $m$-dimensional domains ($m \ge 1$) employing an $m$-dimensional Cartesian grid. For an $N$-point discretization, the method runs at a computational cost of $\mathcal{O}(N)$ operations for standard integrals and $\mathcal{O}(N \log N)$ operations for convolutions, following, in either case, a one-time $\mathcal{O}(N \log N)$ precomputation step (not required in dimension $m = 1$). The core idea underlying TFF is to approximate the characteristic function of the integration domain by a truncated Fourier expansion of it over a suitably extended periodic domain, and to evaluate the resulting integrals via trapezoidal quadrature on a Cartesian grid with an appropriately chosen discretization size. Despite its conceptual simplicity, TFF attains high-order accuracy even for complex, possibly non-smooth or even non-Lipschitz geometries. A complete theoretical analysis is provided that establishes the superalgebraic convergence (i.e., convergence faster than any negative power of $N$) of the overall approach.
