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，用于高效求解在不确定性下的PDE约束优化中通常出现的大型鞍点方程组。该算法基于一种集体平滑器，每次迭代时遍历计算网格的节点，并求解一个降维鞍点系统，其大小取决于用于离散概率空间的样本数量$N$。我们证明，该系统可以达到最优$O(N)$复杂度的求解效率。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性系统；一个带箱式约束和控制变量$L^1$范数惩罚的非光滑问题，其中多重网格方法用于半光滑牛顿迭代；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。在所有案例中，多重网格算法均表现出良好的性能和对所有感兴趣参数的鲁棒性。