Risk Averse PDE Constrained Optimization Using the Conditional Value At Risk
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 (CVaR) for such PDE constrained optimization problems. These approximations are based on the primal and dual formulations of CVaR. For the primal problem, we introduce a smooth approximation of CVaR in order to utilize derivative based optimization algorithms and to take advantage of the convergence properties of quadrature based discretizations. For this smoothed CVaR, 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.
SIAM J. Optim.