Many problems in engineering and sciences require the solution of large scale optimization constrained by partial differential equations (PDEs). Though PDE-constrained optimization is itself challenging, most applications pose additional complexity, namely, uncertain parameters in the PDEs. Uncertainty quantification (UQ) is necessary to characterize, prioritize, and study the influence of these uncertain parameters. Sensitivity analysis, a classical tool in UQ, is frequently used to study the sensitivity of a model to uncertain parameters. In this article, we introduce "hyper-differential sensitivity analysis" which considers the sensitivity of the solution of a PDE-constrained optimization problem to uncertain parameters. Our approach is a goal-oriented analysis which may be viewed as a tool to complement other UQ methods in the service of decision making and robust design. We formally define hyper-differential sensitivity indices and highlight their relationship to the existing optimization and sensitivity analysis literatures. Assuming the presence of low rank structure in the parameter space, computational efficiency is achieved by leveraging a generalized singular value decomposition in conjunction with a randomized solver which converts the computational bottleneck of the algorithm into an embarrassingly parallel loop. Two multi-physics examples, consisting of nonlinear steady state control and transient linear inversion, demonstrate efficient identification of the uncertain parameters which have the greatest influence on the optimal solution.

中文翻译：

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偏微分方程约束优化中不确定参数的超微分灵敏度分析

工程和科学中的许多问题需要求解受偏微分方程 (PDE) 约束的大规模优化。尽管 PDE 约束优化本身具有挑战性，但大多数应用程序都会带来额外的复杂性，即 PDE 中的不确定参数。不确定性量化 (UQ) 是表征、确定优先级和研究这些不确定参数的影响所必需的。灵敏度分析是 UQ 中的经典工具，经常用于研究模型对不确定参数的敏感性。在本文中，我们介绍了“超微分敏感性分析”，它考虑了 PDE 约束优化问题的解对不确定参数的敏感性。我们的方法是面向目标的分析，可以将其视为在决策和稳健设计服务中补充其他 UQ 方法的工具。我们正式定义了超微分敏感性指数，并强调了它们与现有优化和敏感性分析文献的关系。假设参数空间中存在低秩结构，计算效率是通过利用广义奇异值分解结合随机求解器来实现的，该求解器将算法的计算瓶颈转换为令人尴尬的并行循环。由非线性稳态控制和瞬态线性反演组成的两个多物理场示例证明了对对最优解影响最大的不确定参数的有效识别。