A distributed optimal control problem with the constraint of a parabolic partial differential equation is considered. Boundary value methods are used to solve the coupled initial/final value problems arising from the first order optimality conditions for this problem. We use a block triangular preconditioning strategy for solving the resulting two-by-two linear system. By making use of a matching strategy and a Kronecker product-based splitting technique we establish a Kronecker product-based approximation to the Schur complement. Since the Schur complement approximation is in a form of one Kronecker product structure, the preconditioner can be implemented efficiently. Numerical experiments are presented to illustrate the accuracy and computational efficiency of the proposed approach.

中文翻译：

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抛物线型最优控制问题的边值方法离散化的预处理迭代方法

考虑具有抛物线偏微分方程约束的分布式最优控制问题。边界值方法用于解决由一阶最优性条件引起的耦合初始/最终值问题。我们使用块三角预处理策略来求解所得的二乘二线性系统。通过使用匹配策略和基于Kronecker产品的拆分技术，我们建立了基于Schron补品的基于Kronecker产品的近似值。由于Schur补码近似为一种Kronecker乘积结构的形式，因此可以有效地实现预处理器。数值实验表明了该方法的准确性和计算效率。