  We consider optimal experimental design (OED) for nonlinear inverse problems
within the Bayesian framework. Optimizing the data acquisition process for
large-scale nonlinear Bayesian inverse problems is a computationally
challenging task since the posterior is typically intractable and
commonly-encountered optimality criteria depend on the observed data. Since
these challenges are not present in OED for linear Bayesian inverse problems,
we propose an approach based on first linearizing the associated forward
problem and then optimizing the experimental design. Replacing an accurate but
costly model with some linear surrogate, while justified for certain problems,
can lead to incorrect posteriors and sub-optimal designs if model discrepancy
is ignored. To avoid this, we use the Bayesian approximation error (BAE)
approach to formulate an A-optimal design objective for sensor selection that
is aware of the model error. In line with recent developments, we prove that
this uncertainty-aware objective is independent of the exact choice of
linearization. This key observation facilitates the formulation of an
uncertainty-aware OED objective function using a completely trivial linear map,
the zero map, as a surrogate to the forward dynamics. The base methodology is
also extended to marginalized OED problems, accommodating uncertainties arising
from both linear approximations and unknown auxiliary parameters. Our approach
only requires parameter and data sample pairs, hence it is particularly
well-suited for black box forward models. We demonstrate the effectiveness of
our method for finding optimal designs in an idealized subsurface flow inverse
problem and for tsunami detection.
