We consider optimal experimental design (OED) for nonlinear inverse problems within the Bayesian framework. Optimizing the data acquisition process for large-scale nonlinear Bayesian inverse problems is a computationally challenging task since the posterior is typically intractable and commonly-encountered optimality criteria depend on the observed data. Since these challenges are not present in OED for linear Bayesian inverse problems, we propose an approach based on first linearizing the associated forward problem and then optimizing the experimental design. Replacing an accurate but costly model with some linear surrogate, while justified for certain problems, can lead to incorrect posteriors and sub-optimal designs if model discrepancy is ignored. To avoid this, we use the Bayesian approximation error (BAE) approach to formulate an A-optimal design objective for sensor selection that is aware of the model error. In line with recent developments, we prove that this uncertainty-aware objective is independent of the exact choice of linearization. This key observation facilitates the formulation of an uncertainty-aware OED objective function using a completely trivial linear map, the zero map, as a surrogate to the forward dynamics. The base methodology is also extended to marginalized OED problems, accommodating uncertainties arising from both linear approximations and unknown auxiliary parameters. Our approach only requires parameter and data sample pairs, hence it is particularly well-suited for black box forward models. We demonstrate the effectiveness of our method for finding optimal designs in an idealized subsurface flow inverse problem and for tsunami detection.

翻译：我们考虑贝叶斯框架下非线性反问题的最优实验设计（OED）。对于大规模非线性贝叶斯反问题，优化数据采集过程是一项计算上极具挑战的任务，因为后验分布通常难以处理，且常见的最优性准则依赖于观测数据。由于这些问题在贝叶斯线性反问题的OED中并不存在，我们提出一种方法：先对相关正问题进行线性化，再优化实验设计。用线性替代模型替代精确但高成本的原模型，虽在某些问题中合理，但若忽略模型误差，可能导致错误的后验分布和次优设计。为避免此问题，我们采用贝叶斯近似误差（BAE）方法，为传感器选择构建一个考虑模型误差的A-最优设计目标函数。与最新研究进展一致，我们证明该不确定性感知的目标函数不依赖于线性化的具体选择。这一关键发现使我们能够使用完全平凡的线性映射（零映射）作为正演动力学的替代模型，来构建不确定性感知的OED目标函数。该基础方法还被推广到边缘化OED问题，同时处理由线性近似和未知辅助参数引起的不确定性。我们的方法仅需参数和观测数据的样本对，因此特别适用于黑箱正演模型。我们通过理想化地下流体反问题和海啸检测中的最优设计实例，证明了该方法的有效性。