傅里叶积分算子的快速计算
Fast Computation of Fourier Integral Operators
AI总结：
针对波动传播和双曲方程中出现的振荡积分，提出一种基于极坐标楔形局部化和核分解的快速算法，将计算复杂度降至O(N^{2.5} log N)，存储需求降至O(√N)。
摘要：
我们引入了一种通用算法，用于快速计算与波动传播和一般双曲方程相关的问题中经常出现的某些类型的振荡积分。问题是在笛卡尔网格点上数值计算形如 $\int e^{2πi Φ(x,ξ)} a(x,ξ) \hat{f}(ξ) \mathrm{d}ξ$ 的所谓傅里叶积分算子（FIO），其中 $ξ$ 是频率变量，$\hat f(ξ)$ 是输入 $f$ 的傅里叶变换，$a(x,ξ)$ 是振幅，$Φ(x,ξ)$ 是相位函数，其大小通常与 $|ξ|$ 相当；因此积分在高频时高度振荡。由于 FIO 是一个稠密矩阵，对于大小为 $N \times N$ 的笛卡尔网格上的输入，朴素矩阵向量积需要 $O(N^4)$ 次操作。 本文开发了一种新的数值算法，仅需 $O(N^{2.5} \log N)$ 次操作，存储空间低至 $O(\sqrt{N})$。该算法通过在频率平面上将积分局部化到具有小角孔径的极坐标楔形区域上工作。在每个楔形上，算法将核 $e^{2 πi Φ(x,ξ)} a(x,ξ)$ 分解为两个分量：1) 通过非均匀 FFT 处理的微分同胚，2) 通过空间和频率变量的数值分离处理的残差因子。复杂度和精度估计的关键在于残差核的分离秩可证明与问题规模无关。几个数值例子展示了所提方法的效率和精度。我们还讨论了我们的思想在反射地震学等各种应用中的潜力。
英文摘要：
We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation and general hyperbolic equations. The problem is to evaluate numerically a so-called Fourier integral operator (FIO) of the form $\int e^{2πi Φ(x,ξ)} a(x,ξ) \hat{f}(ξ) \mathrm{d}ξ$ at points given on a Cartesian grid. Here, $ξ$ is a frequency variable, $\hat f(ξ)$ is the Fourier transform of the input $f$, $a(x,ξ)$ is an amplitude and $Φ(x,ξ)$ is a phase function, which is typically as large as $|ξ|$; hence the integral is highly oscillatory at high frequencies. Because an FIO is a dense matrix, a naive matrix vector product with an input given on a Cartesian grid of size $N$ by $N$ would require $O(N^4)$ operations. This paper develops a new numerical algorithm which requires $O(N^{2.5} \log N)$ operations, and as low as $O(\sqrt{N})$ in storage space. It operates by localizing the integral over polar wedges with small angular aperture in the frequency plane. On each wedge, the algorithm factorizes the kernel $e^{2 πi Φ(x,ξ)} a(x,ξ)$ into two components: 1) a diffeomorphism which is handled by means of a nonuniform FFT and 2) a residual factor which is handled by numerical separation of the spatial and frequency variables. The key to the complexity and accuracy estimates is that the separation rank of the residual kernel is \emph{provably independent of the problem size}. Several numerical examples demonstrate the efficiency and accuracy of the proposed methodology. We also discuss the potential of our ideas for various applications such as reflection seismology.
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