  We present an efficient method for computing A-optimal experimental designs
for infinite-dimensional Bayesian linear inverse problems governed by partial
differential equations (PDEs). Specifically, we address the problem of
optimizing the location of sensors (at which observational data are collected)
to minimize the uncertainty in the parameters estimated by solving the inverse
problem, where the uncertainty is expressed by the trace of the posterior
covariance. Computing optimal experimental designs (OEDs) is particularly
challenging for inverse problems governed by computationally expensive PDE
models with infinite-dimensional (or, after discretization, high-dimensional)
parameters. To alleviate the computational cost, we exploit the problem
structure and build a low-rank approximation of the parameter-to-observable
map, preconditioned with the square root of the prior covariance operator. This
relieves our method from expensive PDE solves when evaluating the optimal
experimental design objective function and its derivatives. Moreover, we employ
a randomized trace estimator for efficient evaluation of the OED objective
function. We control the sparsity of the sensor configuration by employing a
sequence of penalty functions that successively approximate the
$\ell_0$-"norm"; this results in binary designs that characterize optimal
sensor locations. We present numerical results for inference of the initial
condition from spatio-temporal observations in a time-dependent
advection-diffusion problem in two and three space dimensions. We find that an
optimal design can be computed at a cost, measured in number of forward PDE
solves, that is independent of the parameter and sensor dimensions. We
demonstrate numerically that $\ell_0$-sparsified experimental designs obtained
via a continuation method outperform $\ell_1$-sparsified designs.
