A rational deferred correction approach to parabolic optimal control problems
A rational deferred correction approach to parabolic optimal control problems
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DOI：
10.1093/imanum/drx046
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发表时间：
2018-10
影响因子：
2.1
通讯作者：
S. Güttel;J. Pearson
中科院分区：
文献类型：
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作者：
S. Güttel;J. Pearson
准确有效地求解瞬态偏微分方程约束优化问题是一项具有挑战性的任务，这在很大程度上是由于需要求解的矩阵系统的维数非常高。我们为瞬态偏微分方程的耦合系统设计了一种新的延迟校正方法，允许迭代地提高低阶时间步进方案的精度。我们考虑我们方法的两种变体，分裂版本和耦合版本，并分析它们的收敛特性。然后，我们在许多 PDE 约束优化问题上测试我们的方法。我们获得的解精度远远优于解决单个离散问题时所达到的精度，特别是在精度受到时间离散化限制的情况下。我们的方法允许直接重用现有的求解器来生成矩阵系统，以及最先进的预处理策略。
The accurate and efficient solution of time-dependent PDE-constrained optimization problems is a challenging task, in large part due to the very high dimension of the matrix systems that need to be solved. We devise a new deferred correction method for coupled systems of time-dependent PDEs, allowing one to iteratively improve the accuracy of low-order time stepping schemes. We consider two variants of our method, a splitting and a coupling version, and analyze their convergence properties. We then test our approach on a number of PDE-constrained optimization problems. We obtain solution accuracies far superior to that achieved when solving a single discretized problem, in particular in cases where the accuracy is limited by the time discretization. Our approach allows for the direct reuse of existing solvers for the resulting matrix systems, as well as state-of-the-art preconditioning strategies.