SIAM Journal on Optimization, Volume 31, Issue 3, Page 1850-1876, January 2021.
An algorithm is proposed to solve robust control problems constrained by partial differential equations with uncertain coefficients, based on the so-called MG/OPT framework. The levels in the MG/OPT hierarchy correspond to discretization levels of the PDE, as usual. For stochastic problems, the relevant quantities (such as the gradient) contain expected value operators on each of these levels. They are estimated using a multilevel Monte Carlo method, the specifics of which depend on the MG/OPT level. Each of the optimization levels then contains multiple underlying multilevel Monte Carlo levels. The MG/OPT hierarchy allows the algorithm to exploit the structure inherent in the PDE, speeding up the convergence to the optimum. In contrast, the multilevel Monte Carlo hierarchy exists to exploit structure present in the stochastic dimensions of the problem. A statement about the asymptotic cost of the algorithm is proven, and some additional properties are discussed. The performance of the algorithm is numerically investigated for three test cases. A reduction in the number of samples required on expensive levels and therefore in computation time can be observed.

中文翻译：

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MG/OPT 和多级蒙特卡罗用于 PDE 的稳健优化

SIAM 优化杂志，第 31 卷，第 3 期，第 1850-1876 页，2021 年 1 月。
基于所谓的MG/OPT框架，提出了一种求解系数不确定的偏微分方程约束的鲁棒控制问题的算法。像往常一样，MG/OPT 层次结构中的级别对应于 PDE 的离散化级别。对于随机问题，相关量（例如梯度）在每个级别上都包含期望值算子。它们是使用多级蒙特卡罗方法估计的，其细节取决于 MG/OPT 级别。然后，每个优化级别都包含多个底层多级蒙特卡罗级别。MG/OPT 层次结构允许算法利用 PDE 中固有的结构，加速收敛到最优。相比之下，多级蒙特卡罗层次结构的存在是为了利用问题的随机维度中存在的结构。证明了关于算法的渐近成本的陈述，并讨论了一些附加属性。针对三个测试用例对算法的性能进行了数值研究。可以观察到在昂贵的水平上所需的样本数量减少，因此计算时间也减少了。