We introduce a new class of multilevel, adaptive, dual-space methods for computing fast convolutional transforms. 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. The main novelty of DMK is that the interaction at each scale is diagonalized by a short Fourier transform, permitting the use of separation of variables, but without requiring the FFT for its asymptotic performance. 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 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.

翻译：我们提出了一类新型多层级自适应双空间方法，用于快速计算卷积变换。该方法适用于广泛类别的核函数，涵盖经典偏微分方程的格林函数、幂函数以及统计学与机器学习中使用的径向基函数等。DMK（双空间多层级核分裂）框架采用网格层级化策略：在最粗层级计算平滑后的相互作用，随后在越来越精细的尺度上进行逐级修正，直至问题完全局域化，最后采用直接求和。DMK的核心创新在于：每个尺度的相互作用通过短傅里叶变换实现对角化，从而允许采用分离变量法，但无需依赖快速傅里叶变换来保证渐近性能。该框架显著简化了快速多极子的算法结构，统一了快速多极子算法、埃瓦尔德求和法与多层级求和法，在完全自适应的场景下仍能达到每网格点计算复杂度与快速傅里叶变换相当的性能。针对连续源分布，通过将最精细层核函数近似为高斯函数之和与高度局域化余项，进一步加速了局部相互作用的计算。高斯卷积通过张量积变换实现，余项则采用渐近方法求解。我们通过二维与三维的大量数值算例，展示了DMK在连续源与离散源场景中的性能表现。