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:本文主要研究采用随机方法数值积分$C^r$函数。我们考虑方差缩减方法,分两步进行。首先,将初始区间划分为小区间,并用基于分割结果的分段多项式插值逼近被积函数。然后,在被积函数和插值函数之差上应用随机逼近。最终的积分逼近值由这两部分之和得到。虽然非自适应(均匀分割加简单蒙特卡罗方法)就已经能达到最优收敛速率,然而特殊的自适应算法能够显著降低积分误差的渐近因子,具体效果取决于积分函数本身。对于具有快速变化$r$阶导数的函数,自适应方法的提高效果是非常显著的,这对实际计算有着重要的意义。此外,提出的自适应方法易于实现并可以用于自动积分。