  Optimization constrained by high-fidelity computational models has potential
for transformative impact. However, such optimization is frequently
unattainable in practice due to the complexity and computational intensity of
the model. An alternative is to optimize a low-fidelity model and use limited
evaluations of the high-fidelity model to assess the quality of the solution.
This article develops a framework to use limited high-fidelity simulations to
update the optimization solution computed using the low-fidelity model.
Building off a previous article [22], which introduced hyper-differential
sensitivity analysis with respect to model discrepancy, this article provides
novel extensions of the algorithm to enable uncertainty quantification of the
optimal solution update via a Bayesian framework. Specifically, we formulate a
Bayesian inverse problem to estimate the model discrepancy and propagate the
posterior model discrepancy distribution through the post-optimality
sensitivity operator for the low-fidelity optimization problem. We provide a
rigorous treatment of the Bayesian formulation, a computationally efficient
algorithm to compute posterior samples, a guide to specify and interpret the
algorithm hyper-parameters, and a demonstration of the approach on three
examples which highlight various types of discrepancy between low and
high-fidelity models.
