Quasi-Monte Carlo methods for optimal control problems subject to PDE constraints under uncertainty
Philipp Guth, University of Mannheim
We study the application of tailored quasi-Monte Carlo (QMC) methods to a class of optimal control problems subject to partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of an elliptic or parabolic PDE with a random thermal diffusion coefficient, steered by a linear 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 with respect to 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.