We consider optimal sensor placement for a family of linear Bayesian inverse problems characterized by a deterministic hyper-parameter. The hyper-parameter describes distinct configurations in which measurements can be taken of the observed physical system. To optimally reduce the uncertainty in the system's model with a single set of sensors, the initial sensor placement needs to account for the non-linear state changes of all admissible configurations. We address this requirement through an observability coefficient which links the posteriors' uncertainties directly to the choice of sensors. We propose a greedy sensor selection algorithm to iteratively improve the observability coefficient for all configurations through orthogonal matching pursuit. The algorithm allows explicitly correlated noise models even for large sets of candidate sensors, and remains computationally efficient for high-dimensional forward models through model order reduction. We demonstrate our approach on a large-scale geophysical model of the Perth Basin, and provide numerical studies regarding optimality and scalability with regard to classic optimal experimental design utility functions.

翻译：我们考虑一类由确定性超参数刻画的线性贝叶斯逆问题的传感器最优布置问题。该超参数描述了观测物理系统可进行测量的不同配置。为通过单一传感器集最优地降低系统模型的不确定性，初始传感器布置需考虑所有可行配置的非线性状态变化。我们通过可观测性系数应对这一需求，该系数将后验不确定性直接关联至传感器选择。我们提出一种贪婪传感器选择算法，通过正交匹配追踪迭代优化所有配置下的可观测性系数。该算法即使在候选传感器集合规模较大时也能处理显式相关噪声模型，并通过模型降阶保持高维正向模型的计算效率。我们在珀斯盆地的大尺度地球物理模型上验证了所提方法，并针对经典最优实验设计效用函数的最优性与可扩展性提供了数值研究。