Inversion for parameters of a physical process typically requires taking expensive measurements, and the task of finding an optimal set of measurements is known as the optimal design problem. Surprisingly, measurement locations in optimal designs are sometimes extremely clustered, and researchers often avoid measurement clusterization by modifying the optimal design problem. We consider a certain flavor of the optimal design problem, based on the Bayesian D-optimality criterion, and suggest an analytically tractable model for D-optimal designs in Bayesian linear inverse problems over Hilbert spaces. We demonstrate that measurement clusterization is a generic property of D-optimal designs, and prove that correlated noise between measurements mitigates clusterization. We also give a full characterization of D-optimal designs under our model: We prove that D-optimal designs uniformly reduce uncertainty in a select subset of prior covariance eigenvectors. Finally, we show how measurement clusterization is a consequence of the characterization mentioned above and the pigeonhole principle.

翻译：反演物理过程的参数通常需要昂贵的测量，寻找最优测量集的任务被称为最优设计问题。令人惊讶的是，最优设计中的测量位置有时会出现极端聚类现象，研究人员常通过修改最优设计问题来避免测量聚类。本文基于贝叶斯D最优性准则，考虑特定形式的最优设计问题，并提出一个解析可处理的模型，用于Hilbert空间上贝叶斯线性反问题的D最优设计。我们证明测量聚类是D最优设计的固有性质，并指出测量间的相关噪声会缓解聚类现象。此外，我们给出该模型下D最优设计的完整刻画：证明D最优设计会均匀地减少先验协方差特征向量某一选定子集的不确定性。最后，我们展示测量聚类如何是上述刻画与鸽巢原理的共同结果。