  We apply the sample average approximation (SAA) method to risk-neutral
optimization problems governed by nonlinear partial differential equations
(PDEs) with random inputs. We analyze the consistency of the SAA optimal values
and SAA solutions. Our analysis exploits problem structure in PDE-constrained
optimization problems, allowing us to construct deterministic, compact subsets
of the feasible set that contain the solutions to the risk-neutral problem and
eventually those to the SAA problems. The construction is used to study the
consistency using results established in the literature on stochastic
programming. The assumptions of our framework are verified on three nonlinear
optimization problems under uncertainty.
