Approximate computational approaches for Bayesian sensor placement in high dimensions

Since the cost of installing and maintaining sensors is usually high, sensor locations should always be strategically selected to extract most of the information. For inferring certain quantities of interest (QoIs) using sensor data, it is desirable to explore the dependency between observables and QoIs to identify optimal placement of sensors. Mutual information is a popular dependency measure, however, its estimation in high dimensions is challenging as it requires a large number of samples. This also comes at a significant computational cost when samples are obtained by simulating complex physics-based models. Similarly, identifying the optimal design/location requires a large number of mutual information evaluations to explore a continuous design space. To address these challenges, two novel approaches are proposed. First, instead of estimating mutual information in high-dimensions, we map the limited number of samples onto a lower dimensional space while capturing dependencies between the QoIs and observables. We then estimate a lower bound of the original mutual information in this low dimensional space, which becomes our new dependence measure between QoIs and observables. Second, we use Bayesian optimization to search for optimal sensor locations in a continuous design space while reducing the number of lower bound evaluations. Numerical results on both synthetic and real data are provided to compare the performance of the lower bound with the estimate of mutual information in high dimensions, and a puff-based dispersion model is used to evaluate the sensor placement of the Bayesian optimization for a chemical release problem. The results show that the proposed approaches are both effective and efficient in capturing dependencies and inferring the QoIs.