We present a combination technique based on mixed differences of both spatial approximations and quadrature formulae for the stochastic variables to solve efficiently a class of Optimal Control Problems (OCPs) constrained by random partial differential equations. The method requires to solve the OCP for several low-fidelity spatial grids and quadrature formulae for the objective functional. All the computed solutions are then linearly combined to get a final approximation which, under suitable regularity assumptions, preserves the same accuracy of fine tensor product approximations, while drastically reducing the computational cost. The combination technique involves only tensor product quadrature formulae, thus the discretized OCPs preserve the convexity of the continuous OCP. Hence, the combination technique avoids the inconveniences of Multilevel Monte Carlo and/or sparse grids approaches, but remains suitable for high dimensional problems. The manuscript presents an a-priori procedure to choose the most important mixed differences and an asymptotic complexity analysis, which states that the asymptotic complexity is exclusively determined by the spatial solver. Numerical experiments validate the results.

翻译：我们提出一种基于空间近似与随机变量求积公式混合差分的组合技术，用于高效求解一类由随机偏微分方程约束的最优控制问题。该方法要求针对多个低保真度空间网格和目标泛函的求积公式求解最优控制问题，随后对所有计算解进行线性组合以获得最终近似解。在适当的正则性假设下，该近似解在保持与精细张量积近似相同精度的同时，大幅降低了计算成本。由于组合技术仅涉及张量积求积公式，因此离散后的最优控制问题保留了连续最优控制问题的凸性。该组合技术避免了多层蒙特卡洛方法及稀疏网格方法的不足，但仍适用于高维问题。本文提出了关键混合差分的先验选择策略，并进行了渐近复杂度分析，表明渐近复杂度完全由空间求解器决定。数值实验验证了上述结论。