Multi-model Monte Carlo methods, such as multi-level Monte Carlo (MLMC) and multifidelity Monte Carlo (MFMC), allow for efficient estimation of the expectation of a quantity of interest given a set of models of varying fidelities. Recently, it was shown that the MLMC and MFMC estimators are both instances of the approximate control variates (ACV) framework [Gorodetsky et al. 2020]. In that same work, it was also shown that hand-tailored ACV estimators could outperform MLMC and MFMC for a variety of model scenarios. Because there is no reason to believe that these hand-tailored estimators are the best among a myriad of possible ACV estimators, a more general approach to estimator construction is pursued in this work. First, a general form of the ACV estimator variance is formulated. Then, the formulation is utilized to generate parametrically-defined estimators. These parametrically-defined estimators allow for an optimization to be pursued over a larger domain of possible ACV estimators. The parametrically-defined estimators are tested on a large set of model scenarios, and it is found that the broader search domain enabled by parametrically-defined estimators leads to greater variance reduction.
翻译:Monte Carlo多模范方法,如多级蒙特卡洛(MLMC)和多纤维蒙特卡洛(MFMC)等多模型方法,能够有效估计对不同忠诚模式的一定利益期望值。最近,显示MLMC和MFMC的估测器是大约控制变数框架[Gorodetsky等人,2020年]的范例。在同一工作中,还显示手工定制的ACV估计仪可以超过MLMC和MFMC的模型假设值。由于没有理由相信这些手工定制的估测器是各种可能的ACV估计器中最好的,因此在这项工作中采用了一种更一般的估测方法。首先,制定了ACV估测器的通用差异表。然后,用这种公式来生成有比对称的估测器。这些经过精确界定的估测仪可以使范围更大的MLMC和MFMC能够对各种模型进行优化,在范围更广的AC测测算器中,在范围上,对AC的测算模型进行更宽的测测测度测测度测算,通过测测测测测测测的模型进行范围的模型进行。