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, 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 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 excellent performances and robustness with respect to the 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PDE约束优化中典型的大规模鞍点方程组。该算法基于一种集体光滑器，每次迭代遍历计算网格节点，并求解一个降阶鞍点系统，其规模取决于离散概率空间所使用的样本数$N$。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：线性二次问题（可能包含局部或边界控制），此时多重网格法直接求解线性最优性系统；带箱式约束和$L^1$范数控制惩罚的非光滑问题，其中多重网格方案用于半光滑牛顿迭代框架内；采用平滑条件风险价值风险度量的风险厌恶问题，此时多重网格法被调用到预处理牛顿迭代中。在所有情况下，该多重网格算法均展现出优异的性能及其对相关参数的鲁棒性。