We show that a very simple randomised algorithm for numerical integration can produce a near optimal rate of convergence for integrals of functions in the $d$-dimensional weighted Korobov space. This algorithm uses a lattice rule with a fixed generating vector and the only random element is the choice of the number of function evaluations. For a given computational budget $n$ of a maximum allowed number of function evaluations, we uniformly pick a prime $p$ in the range $n/2 < p \le n$. We show error bounds for the randomised error, which is defined as the worst case expected error, of the form $O(n^{-\alpha - 1/2 + \delta})$, with $\delta > 0$, for a Korobov space with smoothness $\alpha > 1/2$ and general weights. The implied constant in the bound is dimension-independent given the usual conditions on the weights. We present an algorithm that can construct suitable generating vectors \emph{offline} ahead of time at cost $O(d n^4 / \ln n)$ when the weight parameters defining the Korobov spaces are so-called product weights. For this case, numerical experiments confirm our theory that the new randomised algorithm achieves the near optimal rate of the randomised error.

翻译：我们证明，一种非常简单的数值积分随机算法，对于$d$维加权Korobov空间中函数的积分，能够达到近乎最优的收敛速率。该算法采用具有固定生成向量的格规则，唯一的随机元素是函数求值次数的选择。对于给定的计算预算$n$（最大允许的函数求值次数），我们在区间$n/2 < p \le n$内均匀选取一个素数$p$。我们针对随机误差（定义为最坏情况期望误差）给出了形如$O(n^{-\alpha - 1/2 + \delta})$（其中$\delta > 0$）的误差界，该误差界适用于光滑度$\alpha > 1/2\)且具有一般权重的Korobov空间。在权重的通常条件下，该误差界中的隐含常数与维度无关。我们提出一种算法，当定义Korobov空间的权重参数为所谓的乘积权重时，该算法能够以$O(d n^4 / \ln n)$的计算代价提前离线构造合适的生成向量。针对这种情况，数值实验证实了我们的理论：新的随机算法实现了随机误差的近乎最优速率。