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：连续水平蒙特卡罗是著名的多级蒙特卡罗方法的一种无偏连续版本。其中近似水平被视为连续的，从而生成描述目标量的随机过程。连续水平蒙特卡罗方法自然允许基于逐样本的自适应网格细化，这种细化由目标导向误差估计器指示。在估计器中，逐样本的细化水平通过服从指数分布的随机变量抽取。然而，在实际示例中，这由于样本的高方差而导致更高的计算成本。本文提出了一种连续水平蒙特卡罗的变体，其中利用准蒙特卡罗序列来“采样”指数随机变量。我们为该新型估计器提供了复杂度定理，并表明这在理论和实践中均可降低整个估计器的方差。