Solving optimization problems with transient PDE-constraints is computationally costly due to the number of nonlinear iterations and the cost of solving large-scale KKT matrices. These matrices scale with the size of the spatial discretization times the number of time steps. We propose a new two level domain decomposition preconditioner to solve these linear systems when constrained by the heat equation. Our approach leverages the observation that the Schur-complement is elliptic in time, and thus amenable to classical domain decomposition methods. Further, the application of the preconditioner uses existing time integration routines to facilitate implementation and maximize software reuse. The performance of the preconditioner is examined in an empirical study demonstrating the approach is scalable with respect to the number of time steps and subdomains.

翻译：求解含瞬态偏微分方程约束的优化问题因其非线性迭代次数及大规模KKT矩阵求解的高计算成本而具有挑战性。这类矩阵的规模与空间离散化网格点数乘以时间步数成正比。我们提出了一种新型二层区域分解预处理器，用于求解受热方程约束的线性系统。该方法基于对Schur补在时间维度上具有椭圆性的观察，因此适用于经典区域分解方法。此外，预处理器的应用可借助现有时间积分程序以简化实现并最大化软件复用性。通过实证研究验证了该预处理器的性能，结果表明该方法在时间步数和子区域数量方面具有良好的可扩展性。