This article reviews the application of advanced Monte Carlo techniques in the context of Multilevel Monte Carlo (MLMC). MLMC is a strategy employed to compute expectations which can be biased in some sense, for instance, by using the discretization of a associated probability law. The MLMC approach works with a hierarchy of biased approximations which become progressively more accurate and more expensive. Using a telescoping representation of the most accurate approximation, the method is able to reduce the computational cost for a given level of error versus i.i.d. sampling from this latter approximation. All of these ideas originated for cases where exact sampling from couples in the hierarchy is possible. This article considers the case where such exact sampling is not currently possible. We consider Markov chain Monte Carlo and sequential Monte Carlo methods which have been introduced in the literature and we describe different strategies which facilitate the application of MLMC within these methods.

中文翻译：

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高级多级蒙特卡罗方法

本文回顾了高级蒙特卡罗技术在多级蒙特卡罗 (MLMC) 环境中的应用。MLMC 是一种用于计算在某种意义上可能存在偏差的期望的策略，例如，通过使用相关概率定律的离散化。MLMC 方法使用有偏近似的层次结构，这些近似值逐渐变得更准确和更昂贵。使用最准确近似的伸缩表示，该方法能够降低给定误差水平的计算成本，而不是来自后一种近似的 iid 采样。所有这些想法都源于可以从层次结构中的夫妇中进行精确采样的情况。本文考虑了目前无法进行这种精确采样的情况。