We construct an algebraic multigrid (AMG) based preconditioner for the reduced Hessian of a linear‐quadratic optimization problem constrained by an elliptic partial differential equation. While the preconditioner generalizes a geometric multigrid preconditioner introduced in earlier works, its construction relies entirely on a standard AMG infrastructure built for solving the forward elliptic equation, thus allowing for it to be implemented using a variety of AMG methods and standard packages. Our analysis establishes a clear connection between the quality of the preconditioner and the AMG method used. The proposed strategy has a broad and robust applicability to problems with unstructured grids, complex geometry, and varying coefficients. The method is implemented using the Hypre package and several numerical examples are presented.

中文翻译：

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偏微分方程约束的优化中的Hessian代数多网格预处理

我们构造了一个基于代数多重网格（AMG）的预处理器，用于求解椭圆偏微分方程约束的线性二次优化问题的简化Hessian。虽然预处理器可以概括早期工作中引入的几何多重网格预处理器，但其构建完全依赖于为求解正向椭圆方程而构建的标准AMG基础结构，因此可以使用多种AMG方法和标准软件包来实现。我们的分析在预处理器的质量和所使用的AMG方法之间建立了明确的联系。所提出的策略对于非结构化网格，复杂的几何形状和变化的系数具有广泛而强大的适用性。该方法使用Hypre软件包实现，并给出了几个数值示例。