We propose a control-oriented optimal experimental design (cOED) approach for linear PDE-constrained Bayesian inverse problems. In particular, we consider optimal control problems with uncertain parameters that need to be estimated by solving an inverse problem, which in turn requires measurement data. We consider the case where data is collected at a set of sensors. While classical Bayesian OED techniques provide experimental designs (sensor placements) that minimize the posterior uncertainty in the inversion parameter, these designs are not tailored to the demands of the optimal control problem. In the present control-oriented setting, we prioritize the designs that minimize the uncertainty in the state variable being controlled or the control objective. We propose a mathematical framework for uncertainty quantification and cOED for parameterized PDE-constrained optimal control problems with linear dependence to the control variable and the inversion parameter. We also present scalable computational methods for computing control-oriented sensor placements and for quantifying the uncertainty in the control objective. Additionally, we present illustrative numerical results in the context of a model problem motivated by heat transfer applications.

翻译：针对线性偏微分方程约束的贝叶斯反问题，我们提出了一种面向控制的最优实验设计方法。具体而言，我们研究参数不确定的最优控制问题，这些参数需要通过求解反问题来估计，而反问题本身又依赖于测量数据。我们考虑在传感器网络中采集数据的情形。传统的贝叶斯最优实验设计方法提供的实验方案（传感器布局）旨在最小化反演参数的后验不确定性，但这类设计并未针对最优控制问题的需求进行优化。在本研究面向控制的框架下，我们优先考虑能够最小化受控状态变量或控制目标不确定性的设计方案。针对控制变量和反演参数呈线性依赖关系的参数化偏微分方程约束最优控制问题，我们提出了不确定性量化与面向控制最优实验设计的数学框架。同时，我们提出了可扩展的计算方法，用于计算面向控制的传感器布局并量化控制目标的不确定性。此外，我们通过热传导应用背景下的模型问题展示了说明性的数值计算结果。