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We consider the numerical approximation of an 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 squared L2 misfit 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 (OCP) we consider a Finite Element discretization of the underlying PDE, a Monte Carlo sampling method, and gradient-type iterations to obtain the approximate optimal control. We provide full error and complexity analyses of the proposed numerical schemes. In particular we investigate the complexity of a conjugate gradient method applied to the fully discretized OCP (so called Sample Average Approximation), in which the Finite Element discretization and Monte Carlo sample are chosen in advance and kept fixed over the iterations. This is compared with a Stochastic Gradient method on a fixed or varying Finite Element discretization, in which the expectation in the computation of the steepest descent direction is approximated by Monte Carlo estimators, independent across iterations, with small sample sizes. We show in particular that the second strategy results in an improved computational complexity. The theoretical error estimates and complexity results are confirmed by numerical experiments. [ABSTRACT FROM AUTHOR]

We consider the inference problem for parameters in stochastic differential equation (SDE) models from discrete time observations (e.g. experimental or simulation data). Specifically, we study the case where one does not have access to observations of the model itself, but only to a perturbed version that converges weakly to the solution of the model. Motivated by this perturbation argument, we study the convergence of estimation procedures from a numerical analysis point of view. More precisely, we introduce appropriate consistency, stability, and convergence concepts and study their connection. It turns out that standard statistical techniques, such as the maximum likelihood estimator, are not convergent methodologies in this setting, since they fail to be stable. Due to this shortcoming, we introduce and analyse a novel inference procedure for parameters in SDE models which turns out to be convergent. As such, the method is particularly suited for the estimation of parameters in effective (i.e. coarse-grained) models from observations of the corresponding multiscale process. We illustrate these theoretical findings via several numerical examples. [ABSTRACT FROM AUTHOR]