Analysis of stochastic gradient methods for PDE-constrained optimal Control Problems with uncertain parameters

We consider the numerical approximation of a risk-averse optimal control problem for an elliptic partial differential equation (PDE) with random coefficients. Specifically, the control function is a deterministic, distributed forcing term that minimizes the expected mean squared distance between the state (i.e. solution to the PDE) and a target function, subject to a regularization for well posedness. For the numerical treatment of this risk-averse optimal control problem, we consider a Finite Element discretization of the underlying PDEs, a Monte Carlo sampling method, and gradient type iterations to obtain the approximate optimal control. We provide full error and complexity analysis of the proposed numerical schemes. In particular we compare the complexity of a fixed Monte Carlo gradient method, in which the Finite Element discretization and Monte Carlo sample are chosen initially and kept fixed over the gradient iterations, with a Stochastic Gradient method in which the expectation in the computation of the steepest descent direction is approximated by independent Monte Carlo estimators with small sample sizes and possibly varying Finite Element mesh sizes across iterations. We show in particular that the second strategy results in an improved computational complexity. The theoretical error estimates and complexity results are confirmed by our numerical experiments.