面向伪谱数值模拟的高效球谐变换
Efficient Spherical Harmonic Transforms aimed at pseudo-spectral numerical simulations
AI总结：
本文通过利用球谐变换的特性和现代计算机能力，实现了基于Gauss-Legendre求积的高效矢量化算法，显著降低了内存使用和计算时间。
摘要：
在本文中，我们报告了非常高效的球谐变换（SHT）算法。讨论并实现了基于Gauss-Legendre求积的显式矢量化算法变体，并将其集成到包含标量和矢量变换的SHTns库中。主要突破在于通过利用SHT的特定属性以及当前和未来计算机的高级能力，实现了勒让德伴随函数的高效即时计算，即使对于非常高的分辨率也是如此。这使我们能够同时显著减少SHT的内存使用和计算时间。我们测量了算法的性能和精度。尽管SHTns中实现的算法复杂度为$O(N^3)$（其中N是变换的最大谐波度），但它们的性能远优于任何第三方实现，包括低复杂度算法，即使对于高达N=1023的截断也是如此。SHTns作为开源软件可在https://bitbucket.org/nschaeff/shtns获取。
英文摘要：
In this paper, we report on very efficient algorithms for the spherical harmonic transform (SHT). Explicitly vectorized variations of the algorithm based on the Gauss-Legendre quadrature are discussed and implemented in the SHTns library which includes scalar and vector transforms. The main breakthrough is to achieve very efficient on-the-fly computations of the Legendre associated functions, even for very high resolutions, by taking advantage of the specific properties of the SHT and the advanced capabilities of current and future computers. This allows us to simultaneously and significantly reduce memory usage and computation time of the SHT. We measure the performance and accuracy of our algorithms. Even though the complexity of the algorithms implemented in SHTns are in $O(N^3)$ (where N is the maximum harmonic degree of the transform), they perform much better than any third party implementation, including lower complexity algorithms, even for truncations as high as N=1023. SHTns is available at https://bitbucket.org/nschaeff/shtns as open source software.
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