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

Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration