  Optimization is a key tool for scientific and engineering applications,
however, in the presence of models affected by uncertainty, the optimization
formulation needs to be extended to consider statistics of the quantity of
interest. Optimization under uncertainty (OUU) deals with this endeavor and
requires uncertainty quantification analyses at several design locations. The
cost of OUU is proportional to the cost of performing a forward uncertainty
analysis at each design location visited, which makes the computational burden
too high for high-fidelity simulations with significant computational cost.
From a high-level standpoint, an OUU workflow typically has two main
components: an inner loop strategy for the computation of statistics of the
quantity of interest, and an outer loop optimization strategy tasked with
finding the optimal design, given a merit function based on the inner loop
statistics. In this work, we propose to alleviate the cost of the inner loop
uncertainty analysis by leveraging the so-called Multilevel Monte Carlo (MLMC)
method. MLMC has the potential of drastically reducing the computational cost
by allocating resources over multiple models with varying accuracy and cost.
The resource allocation problem in MLMC is formulated by minimizing the
computational cost given a target variance for the estimator. We consider MLMC
estimators for statistics usually employed in OUU workflows and solve the
corresponding allocation problem. For the outer loop, we consider a
derivative-free optimization strategy implemented in the SNOWPAC library; our
novel strategy is implemented and released in the Dakota software toolkit. We
discuss several numerical test cases to showcase the features and performance
of our novel approach with respect to the single fidelity counterpart, based on
standard Monte Carlo evaluation of statistics.
