We present a multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty. The 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 depends on 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. We test the multigrid method on three problems: a linear-quadratic problem 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 within a semismooth Newton iteration; a risk-adverse 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 very good performances and robustness with respect to all parameters of interest.

翻译：我们提出了一种多重网格算法，用于高效求解不确定条件下偏微分方程约束优化中通常出现的大型鞍点方程组。该算法基于集体平滑器，每次迭代时遍历计算网格节点，并求解一个降阶鞍点系统，其规模取决于用于离散概率空间的样本数量$N$。我们证明该降阶系统能以最优$O(N)$复杂度求解。我们在三个问题上测试了该多重网格方法：一个线性二次问题，其中多重网格方法直接用于求解线性最优系统；一个带有箱型约束和控制变量$L^1$范数惩罚的非光滑问题，其中多重网格方案用于半光滑牛顿迭代中；一个采用平滑CVaR风险度量的风险规避问题，其中多重网格方法在预处理牛顿迭代中被调用。在所有情况下，该多重网格算法均表现出优异的性能和关于所有感兴趣参数的鲁棒性。