论文标题
$ c^r $功能降低自适应差异的蒙特卡洛整合：渐近分析
Monte Carlo integration of $C^r$ functions with adaptive variance reduction: an asymptotic analysis
论文作者
论文摘要
本文的主题是使用随机方法的$ c^r $函数的数值集成。我们考虑降低差异方法，这些方法由两个步骤组成。首先，将初始间隔分隔为子间隔，并通过基于获得的分区的分段多项式插值来近似积分。然后，将随机近似应用于积分及其插值的差异。积分的最终近似值是两者的总和。最佳收敛速率已经通过统一（非自适应）分区以及粗蒙特卡洛（Carlo）实现。但是，特殊的自适应技术可以大大降低渐近因子，具体取决于整合体。与非适应性方法相比，这种改进可能是巨大的，特别是对于迅速变化的$ r $ r $ r $ $ r $ r $ r $ r $ r $的功能，这对实际计算具有严重的影响。此外，提出的自适应方法很容易实现，并且可以很好地用于自动集成。
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.
