  We study the application of a tailored quasi-Monte Carlo (QMC) method to a
class of optimal control problems subject to parabolic partial differential
equation (PDE) constraints under uncertainty: the state in our setting is the
solution of a parabolic PDE with a random thermal diffusion coefficient,
steered by a control function. To account for the presence of uncertainty in
the optimal control problem, the objective function is composed with a risk
measure. We focus on two risk measures, both involving high-dimensional
integrals over the stochastic variables: the expected value and the (nonlinear)
entropic risk measure. The high-dimensional integrals are computed numerically
using specially designed QMC methods and, under moderate assumptions on the
input random field, the error rate is shown to be essentially linear,
independently of the stochastic dimension of the problem -- and thereby
superior to ordinary Monte Carlo methods. Numerical results demonstrate the
effectiveness of our method.
