Numerical Performance of Optimized Frolov Lattices in Tensor Product Reproducing Kernel Sobolev Spaces
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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.