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. The cost of OUU is proportional to the cost of performing a forward uncertainty analysis at each design location visited, which makes the computational burden too high for high-fidelity simulations with significant computational cost. From a high-level standpoint, an OUU workflow typically 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. MLMC has the potential of drastically reducing the computational cost by allocating 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 novel approach with respect to the single fidelity counterpart, based on standard Monte Carlo evaluation of statistics.

翻译：优化是科学和工程应用中的关键工具，然而，在受不确定性影响的模型存在的情况下，需要扩展优化框架以考虑目标量的统计特性。不确定性优化（OUU）正是应对这一挑战，要求在多个设计位置开展不确定性量化分析。OUU的计算成本与在每个访问的设计位置上执行正向不确定性分析的成本成正比，这使得对于具有显著计算成本的高保真仿真而言，计算负担过高。从宏观视角看，OUU工作流程通常包含两个核心组成部分：用于计算目标量统计特性的内循环策略，以及基于内循环统计量构建的效能函数，负责寻找最优设计的外循环优化策略。本研究提出通过利用所谓的多层级蒙特卡洛（MLMC）方法来降低内循环不确定性分析的成本。MLMC能够通过将计算资源分配给具有不同精度和成本的多个模型，从而大幅降低计算成本。MLMC中的资源分配问题通过最小化在给定估计量目标方差下的计算成本来构建。我们针对OUU工作流程中常用的统计量设计了MLMC估计量，并求解了相应的分配问题。在外循环方面，我们采用SNOWPAC库中实现的无导数优化策略；这一新策略已在Dakota软件工具包中实现并发布。我们通过多个数值测试案例，展示了本方法相对于基于标准蒙特卡洛统计评估的单保真度方法的特性与性能。