The theme of the present paper is numerical integration of $C^r$ functions using randomized methods. We consider variance reduction methods that consist in two steps. First the initial interval is partitioned into subintervals and the integrand is approximated by a piecewise polynomial interpolant that is based on the obtained partition. Then a randomized approximation is applied on the difference of the integrand and its interpolant. The final approximation of the integral is the sum of both. The optimal convergence rate is already achieved by uniform (nonadaptive) partition plus the crude Monte Carlo; however, special adaptive techniques can substantially lower the asymptotic factor depending on the integrand. The improvement can be huge in comparison to the nonadaptive method, especially for functions with rapidly varying $r$th derivatives, which has serious implications for practical computations. In addition, the proposed adaptive methods are easily implementable and can be well used for automatic integration.

翻译：本文主题为使用随机化方法对$C^r$函数进行数值积分。我们考虑由两个步骤组成的方差缩减方法：首先将初始区间划分为子区间，并基于所得划分构建分段多项式插值函数近似被积函数；随后对原始被积函数与插值函数之差应用随机化逼近。积分最终近似值为两者之和。尽管均匀（非自适应）划分结合原始蒙特卡洛方法已能实现最优收敛速率，但特定自适应技术可显著降低依赖于被积函数的渐近因子。尤其对于$r$阶导数快速变化的函数，相较于非自适应方法，该改进可能极为显著——这对实际计算具有重要影响。此外，所提出的自适应方法易于实现且可广泛应用于自动积分。