Forward simulation-based uncertainty quantification that studies the distribution of quantities of interest (QoI) is a crucial component for computationally robust engineering design and prediction. There is a large body of literature devoted to accurately assessing statistics of QoIs, and in particular, multilevel or multifidelity approaches are known to be effective, leveraging cost-accuracy tradeoffs between a given ensemble of models. However, effective algorithms that can estimate the full distribution of QoIs are still under active development. In this paper, we introduce a general multifidelity framework for estimating the cumulative distribution function (CDF) of a vector-valued QoI associated with a high-fidelity model under a budget constraint. Given a family of appropriate control variates obtained from lower-fidelity surrogates, our framework involves identifying the most cost-effective model subset and then using it to build an approximate control variates estimator for the target CDF. We instantiate the framework by constructing a family of control variates using intermediate linear approximators and rigorously analyze the corresponding algorithm. Our analysis reveals that the resulting CDF estimator is uniformly consistent and asymptotically optimal as the budget tends to infinity, with only mild moment and regularity assumptions on the joint distribution of QoIs. The approach provides a robust multifidelity CDF estimator that is adaptive to the available budget, does not require \textit{a priori} knowledge of cross-model statistics or model hierarchy, and applies to multiple dimensions. We demonstrate the efficiency and robustness of the approach using test examples of parametric PDEs and stochastic differential equations including both academic instances and more challenging engineering problems.

翻译：基于前向模拟的不确定性量化——研究关注量（QoI）的分布——是计算稳健工程设计与预测的关键组成部分。已有大量文献致力于精确评估QoI的统计量，尤其是多层或多保真度方法通过利用给定模型集成间的成本-精度权衡，被证实是有效的。然而，能够估计QoI完整分布的有效算法仍在积极开发中。本文提出了一种通用的多保真度框架，用于在预算约束下估计与高保真度模型关联的向量值QoI的累积分布函数（CDF）。给定一组从低保真度替代模型获得的适当控制变量，该框架首先识别最具成本效益的模型子集，然后利用其构建目标CDF的近似控制变量估计器。我们通过使用中间线性近似器构建一组控制变量来实例化该框架，并严格分析相应算法。分析表明，当预算趋于无穷时，所得CDF估计量在QoI联合分布仅需满足温和矩假设与正则性条件下，具有一致相合性和渐近最优性。该方法能根据可用预算自适应调整，无需先验知道跨模型统计量或模型层次结构，且适用于多维情况。我们通过参数化偏微分方程和随机微分方程的测试算例（包括学术实例及更具挑战性的工程问题）验证了该方法的效率与鲁棒性。