In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence analysis of collective smoothers and collective two-level methods. The multigrid algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size is proportional to the number $N$ of samples used to discretized the probability space. We show that this reduced system can be solved with optimal $O(N)$ complexity. The multigrid method is tested both as a stationary method and as a preconditioner for GMRES on three problems: a linear-quadratic problem, possibly with a local or a boundary control, for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and $L^1$-norm penalization on the control, in which the multigrid scheme is used as an inner solver within a semismooth Newton iteration; a risk-averse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits excellent performances and robustness with respect to the parameters of interest.

翻译：本文提出了一种集体多重网格算法，以高效求解不确定条件下的PDE约束优化中典型出现的大规模鞍点方程组，并发展了集体光滑器和集体双层方法的新收敛性分析。该多重网格算法基于一种集体光滑器，在每次迭代中遍历计算网格节点，并求解一个规模与用于离散概率空间的样本数$N$成比例的简化鞍点系统。我们证明该简化系统可在最优$O(N)$复杂度下求解。该多重网格方法分别作为固定迭代法和GMRES预条件子，在三个问题上进行了测试：线性二次问题（可能包含局部或边界控制），其中多重网格方法直接求解线性最优性系统；具有箱式约束和$L^1$范数控制惩罚的非光滑问题，其中多重网格方案在半光滑牛顿迭代中作为内层求解器；以及使用平滑CVaR风险度量的风险厌恶问题，其中多重网格方法在预条件牛顿迭代中被调用。在所有算例中，该多重网格算法均展现出优异的性能和对关键参数的鲁棒性。