  This work is motivated by the need to study the impact of data uncertainties
and material imperfections on the solution to optimal control problems
constrained by partial differential equations. We consider a pathwise optimal
control problem constrained by a diffusion equation with random coefficient
together with box constraints for the control. For each realization of the
diffusion coefficient we solve an optimal control problem using the variational
discretization [M. Hinze, Comput. Optim. Appl., 30 (2005), pp. 45-61]. Our
framework allows for lognormal coefficients whose realizations are not
uniformly bounded away from zero and infinity. We establish finite element
error bounds for the pathwise optimal controls. This analysis is nontrivial due
to the limited spatial regularity and the lack of uniform ellipticity and
boundedness of the diffusion operator. We apply the error bounds to prove
convergence of a multilevel Monte Carlo estimator for the expected value of the
pathwise optimal controls. In addition we analyze the computational complexity
of the multilevel estimator. We perform numerical experiments in 2D space to
confirm the convergence result and the complexity bound.
