  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.
