Continuous level Monte Carlo is an unbiased, continuous version of the celebrated multilevel Monte Carlo method. The approximation level is assumed to be continuous resulting in a stochastic process describing the quantity of interest. Continuous level Monte Carlo methods allow naturally for samplewise adaptive mesh refinements, which are indicated by goal-oriented error estimators. The samplewise refinement levels are drawn in the estimator from an exponentially-distributed random variable. Unfortunately in practical examples this results in higher costs due to high variance in the samples. In this paper we propose a variant of continuous level Monte Carlo, where a quasi Monte Carlo sequence is utilized to "sample" the exponential random variable. We provide a complexity theorem for this novel estimator and show that this results theoretically and practically in a variance reduction of the whole estimator.

翻译：连续水平蒙特卡洛方法是经典多水平蒙特卡洛方法的一种无偏连续形式。通过将近似水平假设为连续，该方法将关注量描述为随机过程。连续水平蒙特卡洛方法允许自然地实现样本自适应网格细化，此类细化由面向目标的误差估计量指示。样本的自适应细化水平通过指数分布随机变量从估计量中抽取。然而在实际算例中，由于样本方差较高，这种做法会导致计算成本增加。本文提出连续水平蒙特卡洛方法的一种变体，利用拟蒙特卡洛序列"抽样"指数随机变量。我们为该新型估计量建立了复杂度定理，并从理论和实践上证明该方法能够降低整体估计量的方差。