  Optimal experimental design (OED) provides a systematic approach to quantify
and maximize the value of experimental data. Under a Bayesian approach,
conventional OED maximizes the expected information gain (EIG) on model
parameters. However, we are often interested in not the parameters themselves,
but predictive quantities of interest (QoIs) that depend on the parameters in a
nonlinear manner. We present a computational framework of predictive
goal-oriented OED (GO-OED) suitable for nonlinear observation and prediction
models, which seeks the experimental design providing the greatest EIG on the
QoIs. In particular, we propose a nested Monte Carlo estimator for the QoI EIG,
featuring Markov chain Monte Carlo for posterior sampling and kernel density
estimation for evaluating the posterior-predictive density and its
Kullback-Leibler divergence from the prior-predictive. The GO-OED design is
then found by maximizing the EIG over the design space using Bayesian
optimization. We demonstrate the effectiveness of the overall nonlinear GO-OED
method, and illustrate its differences versus conventional non-GO-OED, through
various test problems and an application of sensor placement for source
inversion in a convection-diffusion field.
