The sample average approximation (SAA) approach is applied to risk-neutral
optimization problems governed by semilinear elliptic partial differential
equations with random inputs. After constructing a compact set that contains
the SAA critical points, we derive nonasymptotic sample size estimates for SAA
critical points using the covering number approach. Thereby, we derive upper
bounds on the number of samples needed to obtain accurate critical points of
the risk-neutral PDE-constrained optimization problem through SAA critical
points. We quantify accuracy using expectation and exponential tail bounds.
Numerical illustrations are presented.Comment: 26 pages, 10 figure