实值卷积的混合去混叠与隐式打包
Hybrid Dealiasing and Implicit Packing for Real Convolutions
中文总结 AI 辅助
该研究将混合去混叠扩展到实值卷积，开发两种算法并在FFTW++库中实现高效例程，其性能优于显式零填充方法。
中文摘要 AI 辅助
混合去混叠是一种基于FFT的复值数据线性卷积计算方法，通过隐式执行零填充降低去混叠的成本。我们开发了两种新算法，将混合去混叠扩展到实值卷积：第一种算法利用变换后数据的共轭对称性，直接计算每个残差贡献；第二种算法采用复值混合去混叠，通过一种新的隐式打包技术，将实值数据打包为复值数组，并在变换域中隐式执行所有解包和打包操作。通过递归分解为低维卷积，得到两种算法的多维版本。基于这些算法，在开源FFTW++库中实现了计算等长输入数组卷积的高效例程，这些例程在一维、二维和三维场景下均优于显式零填充的标准方法。在许多情况下，未采用混合去混叠的隐式打包几乎与通用混合去混叠算法一样快，尤其是在多线程场景下。
英文摘要
Hybrid dealiasing is an FFT-based method for computing linear convolutions of complex-valued data that reduces the cost of dealiasing by performing zero padding implicitly. We develop two new algorithms that extend hybrid dealiasing to real-valued convolutions. The first algorithm exploits conjugate symmetries in the transformed data and computes each residue contribution directly. The second algorithm employs complex-valued hybrid dealiasing via a new implicit packing technique, which packs real-valued data into complex-valued arrays and performs all unpacking and packing operations implicitly in the transformed domain. Multidimensional versions of both algorithms are obtained by recursive decomposition into lower-dimensional convolutions. Based on these algorithms, efficient routines to compute convolutions of equal-length input arrays are implemented in the open-source FFTW++ library. These routines outperform the standard method of explicit zero padding in one, two, and three dimensions. In many cases, implicit packing without hybrid dealiasing is nearly as fast as the general hybrid dealiasing algorithm, particularly when multithreading.
