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.

我们研究了定制的准蒙特卡罗（QMC）方法在不确定性下受抛物型偏微分方程（PDE）约束的一类最优控制问题的应用：我们设置中的状态是抛物型 PDE 的解，随机热扩散系数，由控制函数控制。为了考虑最优控制问题中存在的不确定性，目标函数由风险度量组成。我们关注两种风险度量，两者都涉及随机变量的高维积分：期望值和（非线性）熵风险度量。高维积分是使用专门设计的 QMC 方法进行数值计算的，并且在对输入随机场进行适度假设的情况下，误差率基本上是线性的，与问题的随机维度无关，因此优于普通的蒙特卡罗方法。数值结果证明了我们方法的有效性。