标题： 多层蒙特卡洛估计器在不确定性下的无导数优化 显示英文标题

标题： Multilevel Monte Carlo estimators for derivative-free optimization under uncertainty

摘要： 优化是科学和工程应用中的关键工具，然而，在存在受不确定性影响的模型时，优化公式需要扩展以考虑感兴趣量的统计特性。 不确定条件下的优化（OUU）处理这一任务，并且需要在多个设计位置进行不确定性量化分析。 OUU的成本与在每个访问的设计位置执行前向不确定性分析的成本成正比，这使得对于计算成本较高的高保真模拟来说，计算负担过高。 从高层次来看，一个OUU工作流程通常有两个主要组成部分：用于计算感兴趣量的统计特性的内环策略，以及外环优化策略，该策略基于内环统计特性来寻找最优设计。 在这项工作中，我们提出通过利用所谓的多级蒙特卡洛（MLMC）方法来减轻内环不确定性分析的成本。 MLMC有可能通过在具有不同准确度和成本的多个模型上分配资源来极大地减少计算成本。 MLMC中的资源分配问题通过在给定估计器目标方差的情况下最小化计算成本来制定。 我们考虑了OUU工作流程中通常使用的MLMC估计器，并解决了相应的分配问题。 对于外环，我们考虑了在SNOWPAC库中实现的无导数优化策略；我们的新策略已在Dakota软件工具包中实现并发布。 我们讨论了几个数值测试案例，以展示相对于基于标准蒙特卡洛统计评估的单保真度对应方法，我们新方法的特点和性能。 显示英文摘要

摘要： 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.