This manuscript presents a framework for using multilevel quadrature formulae to compute the solution of optimal control problems constrained by random partial differential equations. Our approach consists in solving a sequence of optimal control problems discretized with different levels of accuracy of the physical and probability discretizations. The final approximation of the control is then obtained in a postprocessing step, by suitably combining the adjoint variables computed on the different levels. We present a convergence analysis for an unconstrained linear quadratic problem, and detail our framework for the specific case of a Multilevel Monte Carlo quadrature formula. Numerical experiments confirm the better computational complexity of our MLMC approach compared to a standard Monte Carlo sample average approximation, even beyond the theoretical assumptions.

翻译：本文提出了一种利用多层求积公式计算随机偏微分方程约束下最优控制问题解的框架。我们的方法通过求解一系列在物理离散化和概率离散化上具有不同精度水平的最优控制问题来实现。随后，在后处理步骤中，通过适当组合在不同层级上计算得到的伴随变量，获得控制变量的最终近似解。我们针对无约束线性二次问题给出了收敛性分析，并详细阐述了该方法在多层蒙特卡洛求积公式这一具体情形下的应用框架。数值实验证实，即使在超出理论假设的条件下，我们的多层蒙特卡洛方法相较于标准的蒙特卡洛样本平均近似仍具有更优的计算复杂度。