  We consider optimal sensor placement for hyper-parameterized linear Bayesian
inverse problems, where the hyper-parameter characterizes nonlinear
flexibilities in the forward model, and is considered for a range of possible
values. This model variability needs to be taken into account for the
experimental design to guarantee that the Bayesian inverse solution is
uniformly informative. In this work we link the numerical stability of the
maximum a posterior point and A-optimal experimental design to an observability
coefficient that directly describes the influence of the chosen sensors. We
propose an algorithm that iteratively chooses the sensor locations to improve
this coefficient and thereby decrease the eigenvalues of the posterior
covariance matrix. This algorithm exploits the structure of the solution
manifold in the hyper-parameter domain via a reduced basis surrogate solution
for computational efficiency. We illustrate our results with a steady-state
thermal conduction problem.
