We introduce a new class of multilevel, adaptive, dual‐space methods for computing fast convolutional transformations. These methods can be applied to a broad class of kernels, from the Green's functions for classical partial differential equations (PDEs) to power functions and radial basis functions such as those used in statistics and machine learning. The DMK (dual‐space multilevel kernel‐splitting) framework uses a hierarchy of grids, computing a smoothed interaction at the coarsest level, followed by a sequence of corrections at finer and finer scales until the problem is entirely local, at which point direct summation is applied. Unlike earlier multilevel summation schemes, DMK exploits the fact that the interaction at each scale is diagonalized by a short Fourier transform, permitting the use of separation of variables, but without relying on the FFT. This requires careful attention to the discretization of the Fourier transform at each spatial scale. Like multilevel summation, we make use of a recursive (telescoping) decomposition of the original kernel into the sum of a smooth far‐field kernel, a sequence of difference kernels, and a residual kernel, which plays a role only in leaf boxes in the adaptive tree. At all higher levels in the grid hierarchy, the interaction kernels are designed to be smooth in both physical and Fourier space, admitting efficient Fourier spectral approximations. The DMK framework substantially simplifies the algorithmic structure of the fast multipole method (FMM) and unifies the FMM, Ewald summation, and multilevel summation, achieving speeds comparable to the FFT in work per gridpoint, even in a fully adaptive context. For continuous source distributions, the evaluation of local interactions is further accelerated by approximating the kernel at the finest level as a sum of Gaussians (SOG) with a highly localized remainder. The Gaussian convolutions are calculated using tensor product transforms, and the remainder term is calculated using asymptotic methods. We illustrate the performance of DMK for both continuous and discrete sources with extensive numerical examples in two and three dimensions.

中文翻译：

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用于离散和连续卷积的双空间多级核分裂框架


我们引入了一类新的多级、自适应、双空间方法，用于计算快速卷积变换。这些方法可以应用于广泛的内核，从经典偏微分方程 （PDE） 的格林函数到幂函数和径向基函数，例如统计学和机器学习中使用的函数。DMK （dual-space multilevel kernel‐splitting） 框架使用网格层次结构，在最粗略的级别计算平滑交互，然后在越来越精细的尺度上进行一系列校正，直到问题完全局部化，此时应用直接求和。与早期的多级求和方案不同，DMK 利用了每个尺度上的交互都通过简短的傅里叶变换对角化这一事实，允许使用变量分离，但不依赖于 FFT。这需要仔细注意每个空间尺度上傅里叶变换的离散化。与多级求和一样，我们利用原始内核的递归（伸缩）分解为平滑远场内核、差分内核序列和残差内核的总和，残差内核仅在自适应树的叶框中发挥作用。在网格层次结构的所有更高级别上，交互内核被设计为在物理空间和傅里叶空间中都是平滑的，允许高效的傅里叶频谱近似。DMK 框架大大简化了快速多极点方法 （FMM） 的算法结构，并统一了 FMM、Ewald 求和多级求和，即使在完全自适应的环境中，在每个网格点的工作中也能实现与 FFT 相当的速度。 对于连续源分布，通过将最精细级别的内核近似为具有高度局部余数的高斯 （SOG） 之和，进一步加快了对局部交互的评估。高斯卷积是使用张量积变换计算的，余数项是使用渐近方法计算的。我们通过二维和三维的大量数值示例说明了 DMK 在连续和离散源下的性能。