  We present a method for computing A-optimal sensor placements for
infinite-dimensional Bayesian linear inverse problems governed by PDEs with
irreducible model uncertainties. Here, irreducible uncertainties refers to
uncertainties in the model that exist in addition to the parameters in the
inverse problem, and that cannot be reduced through observations. Specifically,
given a statistical distribution for the model uncertainties, we compute the
optimal design that minimizes the expected value of the posterior covariance
trace. The expected value is discretized using Monte Carlo leading to an
objective function consisting of a sum of trace operators and a binary-inducing
penalty. Minimization of this objective requires a large number of PDE solves
in each step. To make this problem computationally tractable, we construct a
composite low-rank basis using a randomized range finder algorithm to eliminate
forward and adjoint PDE solves. We also present a novel formulation of the
A-optimal design objective that requires the trace of an operator in the
observation rather than the parameter space. The binary structure is enforced
using a weighted regularized $\ell_0$-sparsification approach. We present
numerical results for inference of the initial condition in a subsurface flow
problem with inherent uncertainty in the flow fields and in the initial times.
