Derandomised lattice rules for high dimensional integration

We seek shifted lattice rules that are good for high dimensional integration over the unit cube in the setting of an unanchored weighted Sobolev space of functions with square-integrable mixed first derivatives. Many existing studies rely on random shifting of the lattice, whereas here we work with lattice rules with a deterministic shift. Specifically, we consider 'half-shifted' rules in which each component of the shift is an odd multiple of \(1/(2N)\) where \(N\) is the number of points in the lattice. By applying the principle that there is always at least one choice as good as the average, we show that for a given generating vector there exists a half-shifted rule whose squared worst-case error differs from the shift-averaged squared worst-case error by a term of only order \({1/N^2}\). We carry out numerical experiments where the generating vector is chosen component-by-component (CBC), as for randomly shifted lattices, and where the shift is chosen by a new `CBC for shift' algorithm. The numerical results are encouraging. References J. Dick, F. Y. Kuo, and I. H. Sloan. High-dimensional integration: The quasi-Monte Carlo way. Acta Numer., 22:133–288, 2013. doi:10.1017/S0962492913000044. J. Dick, D. Nuyens, and F. Pillichshammer. Lattice rules for nonperiodic smooth integrands. Numer. Math., 126(2):259–291, 2014. doi:10.1007/s00211-013-0566-0. T. Goda, K. Suzuki, and T. Yoshiki. Lattice rules in non-periodic subspaces of sobolev spaces. Numer. Math., 141(2):399–427, 2019. doi:10.1007/s00211-018-1003-1. F. Y. Kuo. Lattice rule generating vectors. URL http://web.maths.unsw.edu.au/ fkuo/lattice/index.html. D. Nuyens and R. Cools. Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comput., 75:903–920, 2006. doi:10.1090/S0025-5718-06-01785-6. I. H. Sloan and S. Joe. Lattice methods for multiple integration. Oxford Science Publications. Clarendon Press and Oxford University Press, 1994. URL https://global.oup.com/academic/product/lattice-methods-for-multiple-integration-9780198534723. I. H. Sloan and H. Wozniakowski. When are quasi-Monte Carlo algorithms efficient for high dimensional integrals? J. Complex., 14(1):1–33, 1998. doi:10.1006/jcom.1997.0463. I. H. Sloan, F. Y. Kuo, and S. Joe. On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces. Math. Comput., 71:1609–1641, 2002. doi:10.1090/S0025-5718-02-01420-5.

Richardson extrapolation allows truncation of higher-order digital nets and sequences

We study numerical integration of smooth functions defined over the $s$-dimensional unit cube. A recent work by Dick et al. (2019, Richardson extrapolation of polynomial lattice rules. SIAM J. Numer. Anal., 57, 44–69) has introduced so-called extrapolated polynomial lattice rules, which achieve the almost optimal rate of convergence for numerical integration, and can be constructed by the fast component-by-component search algorithm with smaller computational costs as compared to interlaced polynomial lattice rules. In this paper we prove that, instead of polynomial lattice point sets, truncated higher-order digital nets and sequences can be used within the same algorithmic framework to explicitly construct good quadrature rules achieving the almost optimal rate of convergence. The major advantage of our new approach compared to original higher-order digital nets is that we can significantly reduce the precision of points, i.e., the number of digits necessary to describe each quadrature node. This finding has a practically useful implication when either the number of points or the smoothness parameter is so large that original higher-order digital nets require more than the available finite-precision floating-point representations.