SIAM Review, Volume 66, Issue 2, Page 285-285, May 2024.
The Gauss transform---convolution with a Gaussian in the continuous case and the sum of $N$ Gaussians at $M$ points in the discrete case---is ubiquitous in applied mathematics, from solving ordinary and partial differential equations to probability density estimation to science applications in astrophysics, image processing, quantum mechanics, and beyond. For the discrete case, the fast Gauss transform (FGT) enables the approximate calculation of the sum of $N$ Gaussians at $M$ points in order $N + M$ (instead of $NM$) operations by a fast summation strategy, which shares work between the sums at different points, similarly to the fast multipole method. In this issue's Research Spotlights section, “A New Version of the Adaptive Fast Gauss Transform for Discrete and Continuous Sources,” authors Leslie F. Greengard, Shidong Jiang, Manas Rachh, and Jun Wang present a new FGT technique that avoids the use of Hermite and local expansions. The new technique employs Fourier spectral approximations, which are accelerated by nonuniform fast Fourier transforms, and results in a considerably more efficient adaptive implementation. Adaptivity is especially vital for realizing the acceleration from a fast transform when points are highly nonuniform. The paper presents compelling illustrations and examples of the computational approach and the adaptive tree-based hierarchy employed. This hierarchy is used to resolve point distributions down to a refinement level determined by accuracy demands; this results in significantly better work per grid point than conventional FGT techniques. Consequently, the authors note that there are potential key benefits in parallelization of the proposed technique. In addition to the technique's clever composition of a broad variety of advanced computing paradigms and exploitation of mathematical structure to facilitate such fast transforms, the authors present several pathways of future research. For example, the analysis is readily accessible from dimensions larger than the illustrative examples illuminate, and univariate sum-of-exponentials structure also may be exploited; the computing techniques detailed by the authors could be tailored to such regimes. These future directions have broad application in scientific computing.

中文翻译：

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 研究热点


《SIAM 评论》，第 66 卷，第 2 期，第 285-285 页，2024 年 5 月。

高斯变换——连续情况下与高斯卷积以及离散情况下$M$点处的$N$高斯之和——在应用数学中无处不在，从求解常微分方程和偏微分方程到概率密度对天体物理学、图像处理、量子力学等领域的科学应用的估计。对于离散情况，快速高斯变换 (FGT) 可以通过快速求和策略按顺序 $N + M$（而不是 $NM$）运算近似计算 $M$ 点处的 $N$ 高斯之和，它在不同点的总和之间分担工作，类似于快速多极子方法。在本期的研究热点部分“离散和连续源的自适应快速高斯变换的新版本”中，作者 Leslie F. Greengard、Shidong Jiang、Manas Rachh 和 Jun Wang 提出了一种新的 FGT 技术，该技术避免使用 Hermite以及局部扩张。新技术采用傅里叶谱近似，通过非均匀快速傅里叶变换加速，并导致显着更有效的自适应实现。当点高度不均匀时，自适应性对于实现快速变换的加速尤其重要。该论文提供了计算方法和所采用的自适应基于树的层次结构的引人注目的插图和示例。该层次结构用于将点分布解析为由精度要求确定的细化级别；与传统的 FGT 技术相比，这使得每个网格点的工作效果明显更好。因此，作者指出，所提出的技术的并行化具有潜在的关键优势。 除了该技术巧妙地组合了各种先进的计算范式和利用数学结构来促进这种快速转换之外，作者还提出了未来研究的几种途径。例如，可以从比说明性示例所示更大的维度容易地进行分析，并且还可以利用单变量指数和结构；作者详细介绍的计算技术可以针对此类制度进行调整。这些未来的方向在科学计算中具有广泛的应用。