We derive a new parallel-in-time approach for solving large-scale optimization problems constrained by time-dependent partial differential equations arising from fluid dynamics. The solver involves the use of a block circulant approximation of the original matrices, enabling parallelization-in-time via the use of fast Fourier transforms, and we devise bespoke matrix approximations which may be applied within this framework. These make use of permutations, saddle-point approximations, commutator arguments, as well as inner solvers such as the Uzawa method, Chebyshev semi-iteration, and multigrid. Theoretical results underpin our strategy of applying a block circulant strategy, and numerical experiments demonstrate the effectiveness and robustness of our approach on Stokes and Oseen problems. Noteably, satisfying results for the strong and weak scaling of our methods are provided within a fully parallel architecture.

翻译：本文提出了一种新的并行时间方法，用于求解由流体动力学中时间依赖偏微分方程约束的大规模优化问题。该求解器涉及对原始矩阵采用块循环近似，通过快速傅里叶变换实现时间维度上的并行化，并在此框架内设计了可应用的定制矩阵近似方法。这些方法利用了置换、鞍点近似、交换子论证，以及内部求解器如Uzawa方法、Chebyshev半迭代法和多重网格法。理论结果支撑了我们应用块循环策略的方法，数值实验证明了该方法在Stokes和Oseen问题上的有效性与鲁棒性。值得注意的是，在完全并行架构下，我们的方法在强扩展性和弱扩展性方面均取得了令人满意的结果。