We study the numerical integration of functions from isotropic Sobolev spaces $W_p^s([0,1]^d)$ using finitely many function evaluations within randomized algorithms, aiming for the smallest possible probabilistic error guarantee $\varepsilon > 0$ at confidence level $1-\delta \in (0,1)$. For spaces consisting of continuous functions, non-linear Monte Carlo methods with optimal confidence properties have already been known, in few cases even linear methods that succeed in that respect. In this paper we promote a new method called stratified control variates (SCV) and by it show that already linear methods achieve optimal probabilistic error rates in the high smoothness regime without the need to adjust algorithmic parameters to the uncertainty $\delta$. We also analyse a version of SCV in the low smoothness regime where $W_p^s([0,1]^d)$ may contain functions with singularities. Here, we observe a polynomial dependence of the error on $\delta^{-1}$ which cannot be avoided for linear methods. This is worse than what is known to be possible using non-linear algorithms where only a logarithmic dependence on $\delta^{-1}$ occurs if we tune in for a specific value of $\delta$.

翻译：我们研究使用随机算法中有限次函数评估对来自各向同性Sobolev空间$W_p^s([0,1]^d)$的函数进行数值积分，旨在置信水平$1-\delta \in (0,1)$下获得尽可能小的概率误差保证$\varepsilon > 0$。对于由连续函数构成的空间，具有最优置信性质的非线性蒙特卡洛方法已被知晓，在少数情况下，即使是线性方法也能实现这一目标。本文提出一种名为分层控制变量（SCV）的新方法，并证明在高光滑性条件下，线性方法无需根据不确定性$\delta$调整算法参数即可达到最优概率误差率。我们还分析了SCV在低光滑性条件下的变体，其中$W_p^s([0,1]^d)$可能包含具有奇异性的函数。在此情形下，我们观察到误差对$\delta^{-1}$的多项式依赖性，这对于线性方法是不可避免的。这比已知利用非线性算法可能达到的结果更差——若针对特定$\delta$值进行调优，非线性方法仅出现对$\delta^{-1}$的对数依赖性。