In this article, we study preconditioning techniques for the control of the Navier–Stokes equation, where the control only acts on a few parts of the domain. Optimization, discretization, and linearization of the control problem results in a generalized linear saddle‐point system. The Schur complement for the generalized saddle‐point system is very difficult or even impossible to approximate, which prohibits satisfactory performance of the standard block preconditioners. We apply the multilevel sequentially semiseparable (MSSS) preconditioner to the underlying system. Compared with standard block preconditioning techniques, the MSSS preconditioner computes an approximate factorization of the global generalized saddle‐point matrix up to a prescribed accuracy in linear computational complexity. This in turn gives parameter independent convergence for MSSS preconditioned Krylov solvers. We use a simplified wind farm control example to illustrate the performance of the MSSS preconditioner. We also compare the performance of the MSSS preconditioner with the performance of the state‐of‐the‐art preconditioning techniques. Our results show the superiority of the MSSS preconditioning techniques to standard block preconditioning techniques for the control of the Navier–Stokes equation.

中文翻译：

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使用多级顺序半分离矩阵计算对Navier–Stokes控制进行预处理

在本文中，我们研究了用于控制Navier–Stokes方程的预处理技术，其中该控制仅作用于该域的几个部分。控制问题的优化，离散化和线性化导致了广义的线性鞍点系统。广义鞍点系统的Schur补码很难估计甚至无法估计，这妨碍了标准块预处理器的令人满意的性能。我们将多级顺序半分离（MSSS）预处理器应用于基础系统。与标准块预处理技术相比，MSSS预处理器可计算全局广义鞍点矩阵的近似因式分解，线性计算复杂度可达到规定的精度。反过来，这为MSSS预处理的Krylov求解器提供了参数独立的收敛性。我们使用一个简化的风电场控制示例来说明MSSS预处理器的性能。我们还将MSSS预处理器的性能与最先进的预处理技术的性能进行了比较。我们的结果表明，用于控制Navier–Stokes方程的MSSS预处理技术优于标准块预处理技术。