  Optimal experimental design (OED) is the general formalism of sensor
placement and decisions about the data collection strategy for engineered or
natural experiments. This approach is prevalent in many critical fields such as
battery design, numerical weather prediction, geosciences, and environmental
and urban studies. State-of-the-art computational methods for experimental
design, however, do not accommodate correlation structure in observational
errors produced by many expensive-to-operate devices such as X-ray machines or
radar and satellite retrievals. Discarding evident data correlations leads to
biased results, poor data collection decisions, and waste of valuable
resources. We present a general formulation of the OED formalism for
model-constrained large-scale Bayesian linear inverse problems, where
measurement errors are generally correlated. The proposed approach utilizes the
Hadamard product of matrices to formulate the weighted likelihood and is valid
for both finite- and infinite- dimensional Bayesian inverse problems. We also
discuss widely used approaches for relaxation of the binary OED problem, in
light of the proposed pointwise weighting approach, and present a clear
interpretation of the relaxed design and its effect on the observational error
covariance. Extensive numerical experiments are carried out for empirical
verification of the proposed approach by using an advection-diffusion model,
where the objective is to optimally place a small set of sensors, under a
limited budget, to predict the concentration of a contaminant in a bounded
domain.
