  Estimating arbitrary quantities of interest (QoIs) that are non-linear
operators of complex, expensive-to-evaluate, black-box functions is a
challenging problem due to missing domain knowledge and finite budgets.
Bayesian optimal design of experiments (BODE) is a family of methods that
identify an optimal design of experiments (DOE) under different contexts, using
only in a limited number of function evaluations. Under BODE methods,
sequential design of experiments (SDOE) accomplishes this task by selecting an
optimal sequence of experiments while using data-driven probabilistic surrogate
models instead of the expensive black-box function. Probabilistic predictions
from the surrogate model are used to define an information acquisition function
(IAF) which quantifies the marginal value contributed or the expected
information gained by a hypothetical experiment. The next experiment is
selected by maximizing the IAF. A generally applicable IAF is the expected
information gain (EIG) about a QoI as captured by the expectation of the
Kullback-Leibler divergence between the predictive distribution of the QoI
after doing a hypothetical experiment and the current predictive distribution
about the same QoI. We model the underlying information source as a
fully-Bayesian, non-stationary Gaussian process (FBNSGP), and derive an
approximation of the information gain of a hypothetical experiment about an
arbitrary QoI conditional on the hyper-parameters The EIG about the same QoI is
estimated by sample averages to integrate over the posterior of the
hyper-parameters and the potential experimental outcomes. We demonstrate the
performance of our method in four numerical examples and a practical
engineering problem of steel wire manufacturing. The method is compared to two
classic SDOE methods: random sampling and uncertainty sampling.
