  This work introduces a new method to efficiently solve optimization problems
constrained by partial differential equations (PDEs) with uncertain
coefficients. The method leverages two sources of inexactness that trade
accuracy for speed: (1) stochastic collocation based on dimension-adaptive
sparse grids (SGs), which approximates the stochastic objective function with a
limited number of quadrature nodes, and (2) projection-based reduced-order
models (ROMs), which generate efficient approximations to PDE solutions. These
two sources of inexactness lead to inexact objective function and gradient
evaluations, which are managed by a trust-region method that guarantees global
convergence by adaptively refining the sparse grid and reduced-order model
until a proposed error indicator drops below a tolerance specified by
trust-region convergence theory. A key feature of the proposed method is that
the error indicator---which accounts for errors incurred by both the sparse
grid and reduced-order model---must be only an asymptotic error bound, i.e., a
bound that holds up to an arbitrary constant that need not be computed. This
enables the method to be applicable to a wide range of problems, including
those where sharp, computable error bounds are not available; this
distinguishes the proposed method from previous works. Numerical experiments
performed on a model problem from optimal flow control under uncertainty verify
global convergence of the method and demonstrate the method's ability to
outperform previously proposed alternatives.
