Uncertainty is inevitable when solving science and engineering application problems. In the face of
uncertainty, it is essential to determine robust and risk-averse solutions. In this work,
we consider a class of PDE-constrained optimization problems in which the PDE coefficients
and inputs may be uncertain. We introduce two approximations for minimizing the
conditional value-at-risk for such PDE-constrained optimization problems. These approximations are based
on the primal and dual formulations of the conditional value-at-risk. For the primal problem,
we introduce a smooth approximation of the conditional value-at-risk in order to utilize
derivative-based optimization algorithms and to take advantage of the convergence properties
of quadrature-based discretizations. For this smoothed conditional value-at-risk, we prove
differentiability as well as consistency of our approximation. For the dual problem, we
regularize the inner maximization problem, rigorously derive optimality conditions, and demonstrate
the consistency of our approximation. Furthermore, we propose a fixed-point iteration that takes
advantage of the structure of the regularized optimality conditions and provides a means of calculating
worst-case probability distributions based on the given probability level. We conclude with numerical
results.