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

TL;DR

This paper investigates the numerical performance of optimized Frolov lattices in tensor product Sobolev spaces, introducing new generating polynomials and algorithms to improve cubature error reduction and error estimation.

Contribution

It presents novel generating polynomials and an enumeration algorithm for Frolov points, enhancing the accuracy and efficiency of cubature in Sobolev spaces with anisotropic smoothness.

Findings

New generating polynomials significantly reduce integration error.

Developed an algorithm for enumerating Frolov points in non-orthogonal lattices.

Explicit formulas for reproducing kernels enable simulation of worst-case errors.

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.