In this paper, we deal with several aspects of the universal Frolov cubature method, which is known to achieve optimal asymptotic convergence rates in a broad range of function spaces. Even though every admissible lattice has this favorable asymptotic behavior, there are significant differences concerning the precise numerical behavior of the worst-case error. To this end, we propose new generating polynomials that promise a significant reduction in the integration error compared to the classical polynomials. Moreover, we develop a new algorithm to enumerate the Frolov points from non-orthogonal lattices for numerical cubature in the d-dimensional unit cube \([0,1]^d\). Finally, we study Sobolev spaces with anisotropic mixed smoothness and compact support in \([0,1]^d\) and derive explicit formulas for their reproducing kernels. This allows for the simulation of exact worst-case errors which numerically validate our theoretical results.

在本文中，我们讨论了通用Frolov育儿方法的几个方面，已知该方法可在各种功能空间中实现最佳渐近收敛速度。即使每个允许的晶格都具有这种有利的渐近行为，但在最坏情况误差的精确数值行为方面也存在显着差异。为此，我们提出了新的生成多项式，与经典多项式相比，这些多项式有望大大降低积分误差。此外，我们开发了一种新的算法，用于从d维单位立方\（[0,1] ^ d \）的数字正交中从非正交晶格枚举Frolov点。最后，我们研究具有各向异性混合光滑性和\（[[0,1] ^ d \）的紧支撑的Sobolev空间并为其繁殖内核推导明确的公式。这样就可以模拟精确的最坏情况的误差，从而在数值上验证我们的理论结果。