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

The \emph{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 optimising 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\revix{, and marking certain sensors as \emph{dominant} or \emph{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-discretised 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.