Models incorporating uncertain inputs, such as random forces or material
parameters, have been of increasing interest in PDE-constrained optimization.
In this paper, we focus on the efficient numerical minimization of a convex and
smooth tracking-type functional subject to a linear partial differential
equation with random coefficients and box constraints. The approach we take is
based on stochastic approximation where, in place of a true gradient, a
stochastic gradient is chosen using one sample from a known probability
distribution. Feasibility is maintained by performing a projection at each
iteration. In the application of this method to PDE-constrained optimization
under uncertainty, new challenges arise. We observe the discretization error
made by approximating the stochastic gradient using finite elements. Analyzing
the interplay between PDE discretization and stochastic error, we develop a
mesh refinement strategy coupled with decreasing step sizes. Additionally, we
develop a mesh refinement strategy for the modified algorithm using iterate
averaging and larger step sizes. The effectiveness of the approach is
demonstrated numerically for different random field choices
