一种用于含不确定输入的PDE约束优化的多重网格求解器
A multigrid solver for PDE-constrained optimization with uncertain inputs
AI总结：
针对含不确定输入的PDE约束优化中的大型鞍点系统，提出一种基于集体光滑子的多重网格算法，证明其简化系统求解达最优$O(N)$复杂度，在三类问题测试中均表现出优异性能和参数鲁棒性。
摘要：
在本文中，我们提出了一种集体多重网格算法，用于高效求解不确定性下PDE约束优化中通常出现的大型鞍点方程组，并对集体光滑子和集体两层方法开展了全新的收敛性分析。该多重网格算法基于集体光滑子，每次迭代时遍历计算网格的节点，并求解一个规模与概率空间离散化所用样本数$N$成正比的简化鞍点系统。我们证明该简化系统可以最优$O(N)$复杂度求解。\n 该多重网格方法分别作为定常方法和GMRES的预处理器在三类问题上进行了测试：第一类是线性二次问题（可能带有局部控制或边界控制），多重网格方法用于直接求解线性最优性系统；第二类是带有箱型约束和控制量$L^1$范数惩罚的非光滑问题，多重网格格式作为半光滑牛顿迭代中的内求解器使用；第三类是采用平滑CVaR风险度量的风险规避问题，多重网格方法在预处理牛顿迭代中被调用。在所有案例中，该多重网格算法均表现出优异的性能，且对相关参数具有鲁棒性。
英文摘要：
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.
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