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：我们研究从各向同性Sobolev空间$W_p^s([0,1]^d)$中函数数值积分的问题，使用随机算法中有限次函数求值，旨在置信水平$1-\delta \in (0,1)$下实现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，少数情况下甚至存在符合该要求的线性方法。本文提出一种名为分层控制变量法(SCV)的新方法，并据此证明：在高光滑性条件下，仅需线性方法即可实现最优概率误差率，且无需根据不确定性参数$\delta$调整算法参数。我们还分析了$W_p^s([0,1]^d)$可能包含含奇异性函数时低光滑性条件下的SCV变体。在此情形下，我们观察到误差对$\delta^{-1}$存在多项式依赖关系，这对于线性方法而言是不可避免的。这种表现劣于已知使用非线性算法所能达到的效果——若针对特定$\delta$值进行调参，非线性算法仅会出现对$\delta^{-1}$的对数依赖。