Optimization is a key tool for scientific and engineering applications; however, in the presence of models affected by uncertainty, the optimization formulation needs to be extended to consider statistics of the quantity of interest. Optimization under uncertainty (OUU) deals with this endeavor and requires uncertainty quantification analyses at several design locations; i.e., its overall computational cost is proportional to the cost of performing a forward uncertainty analysis at each design location. An OUU workflow has two main components: an inner loop strategy for the computation of statistics of the quantity of interest, and an outer loop optimization strategy tasked with finding the optimal design, given a merit function based on the inner loop statistics. In this work, we propose to alleviate the cost of the inner loop uncertainty analysis by leveraging the so-called multilevel Monte Carlo (MLMC) method, which is able to allocate resources over multiple models with varying accuracy and cost. The resource allocation problem in MLMC is formulated by minimizing the computational cost given a target variance for the estimator. We consider MLMC estimators for statistics usually employed in OUU workflows and solve the corresponding allocation problem. For the outer loop, we consider a derivative-free optimization strategy implemented in the SNOWPAC library; our novel strategy is implemented and released in the Dakota software toolkit. We discuss several numerical test cases to showcase the features and performance of our approach with respect to its Monte Carlo single fidelity counterpart.

中文翻译：

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不确定性下无导数优化的多级蒙特卡洛估计器

优化是科学和工程应用的关键工具；然而，在存在受不确定性影响的模型的情况下，需要扩展优化公式以考虑感兴趣数量的统计数据。不确定性下的优化 (OUU) 致力于解决这一问题，并需要在多个设计位置进行不确定性量化分析；即，其总体计算成本与在每个设计位置执行前向不确定性分析的成本成正比。OUU 工作流程有两个主要组成部分：用于计算感兴趣数量的统计数据的内循环策略，以及负责在给定基于内循环统计数据的评价函数的情况下寻找最佳设计的外循环优化策略。在这项工作中，我们建议通过利用所谓的多级蒙特卡罗（MLMC）方法来减轻内循环不确定性分析的成本，该方法能够以不同的精度和成本在多个模型上分配资源。MLMC 中的资源分配问题是通过最小化给定估计器的目标方差的计算成本来制定的。我们考虑 OUU 工作流程中通常使用的统计 MLMC 估计器并解决相应的分配问题。对于外循环，我们考虑在 SNOWPAC 库中实现的无导数优化策略；我们的新颖策略在 Dakota 软件工具包中实施和发布。我们讨论了几个数值测试用例，以展示我们的方法相对于蒙特卡罗单保真度对应方法的特征和性能。