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arXiv · 1802.08666

Numerical performance of optimized Frolov lattices in tensor product reproducing kernel Sobolev spaces

Christopher Kacwin·Jens Oettershagen·Mario Ullrich·Tino Ullrich
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Abstract

In this paper, we deal with several aspects of the universal Frolov cubature method, that 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 of 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.

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Christopher Kacwin, Jens Oettershagen, Mario Ullrich, Tino Ullrich. 2018-02-23. Numerical performance of optimized Frolov lattices in tensor product reproducing kernel Sobolev spaces. https://arxiv.org/abs/1802.08666

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