基于方向等间距插值的振荡核快速多极子方法
A Directional Equispaced interpolation-based Fast Multipole Method for oscillatory kernels
摘要：
基于振荡Helmholtz核的快速多极子方法（Fast Multipole Method，简称FMF）可以有效地减少求解声学或电磁学中由边界积分方程（BIEs）引起的N体问题的计算量。然而，它们的成本在高频范围内强烈增加。本文介绍了一种新的振荡核方向FMM（defmm - directional equisspaced interpolation based fMM），由于在等间距网格上进行多项式插值，其预计算和应用都得到了FFT加速。我们证明了我们的FFT方法的一致性，并显示如何对称性可以利用在傅立叶域。我们还描述了算法设计的defmm，非常适合的BIE非均匀粒子分布，并提出一个CPU核心上的性能优化。最后，我们表现出重要的性能增益的所有测试情况下，defmm在一个国家的最先进的FMM库振荡内核。
英文摘要：
Fast Multipole Methods (FMMs) based on the oscillatory Helmholtz kernel can reduce the cost of solving N-body problems arising from Boundary Integral Equations (BIEs) in acoustic or electromagnetics. However, their cost strongly increases in the high-frequency regime. This paper introduces a new directional FMM for oscillatory kernels (defmm - directional equispaced interpolation-based fmm), whose precomputation and application are FFT-accelerated due to polynomial interpolations on equispaced grids. We demonstrate the consistency of our FFT approach, and show how symmetries can be exploited in the Fourier domain. We also describe the algorithmic design of defmm, well-suited for the BIE non-uniform particle distributions, and present performance optimizations on one CPU core. Finally, we exhibit important performance gains on all test cases for defmm over a state-of-the-art FMM library for oscillatory kernels.
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