Global optimality conditions for sensor placement, with extensions to binary low-rank A-optimal designs

Abstract The sensor placement problem for stochastic linear inverse problems consists of determining the optimal manner in which sensors can be employed to collect data. Specifically, one wishes to place a limited number of sensors over a large number of candidate locations, quantifying and optimizing over the effect this data collection strategy has on the solution of the inverse problem. In this article, we provide a global optimality condition for the sensor placement problem via a subgradient argument, obtaining sufficient and necessary conditions for optimality, and marking certain sensors as dominant or redundant , i.e. always on or always off. We demonstrate how to take advantage of this optimality criterion to find approximately optimal binary designs, i.e. designs where no fractions of sensors are placed. Leveraging our optimality criteria, we derive a powerful low-rank formulation of the A-optimal design objective for finite element-discretized function space settings, demonstrating its high computational efficiency, particularly in terms of derivatives, and study globally optimal designs for a Helmholtz-type source problem and extensions towards optimal binary designs.