A randomised lattice rule algorithm with pre-determined generating vector and random number of points for Korobov spaces with 0 < α≤ 1/2
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In previous work (Kuo, Nuyens, Wilkes, 2023), we showed that a lattice rule with a pre-determined generating vector but random number of points can achieve the near optimal convergence of O(n^-α-1/2+ϵ), ϵ > 0, for the worst case expected error, or randomised error, for numerical integration of high-dimensional functions in the Korobov space with smoothness α > 1/2. Compared to the optimal deterministic rate of O(n^-α+ϵ), ϵ > 0, such a randomised algorithm is capable of an extra half in the rate of convergence. In this paper, we show that a pre-determined generating vector also exists in the case of 0 < α≤ 1/2. Also here we obtain the near optimal convergence of O(n^-α-1/2+ϵ), ϵ > 0; or in more detail, we obtain O(√(r) n^-α-1/2+1/(2r)+ϵ') which holds for any choices of ϵ' > 0 and r ∈ℕ with r > 1/(2α).
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