Fast iterative solver for the optimal control of time-dependent PDEs with Crank-Nicolson discretization in time
Fast iterative solver for the optimal control of time-dependent PDEs with Crank-Nicolson discretization in time
复制标题
快速迭代求解器，用于通过 Crank-Nicolson 及时离散化对时间相关偏微分方程进行最优控制
DOI：
10.1002/nla.2419
复制
发表时间：
2021
影响因子：
4.3
通讯作者：
Leveque S
中科院分区：
文献类型：
--
作者：
Leveque S
在这篇文章中，我们推导出一种新的，快速的，鲁棒的预处理迭代求解策略，用于一次性解决具有时间依赖的偏微分方程作为约束的最优控制问题，包括热方程和非定常对流扩散方程。在应用优化-然后-离散化方法之后，人们面临由偏微分方程的耦合系统组成的连续一阶最优性条件。与大多数在预处理所得离散化系统中的工作相反，其中（一阶精度）向后欧拉方法用于时间导数的离散化，我们在时间上采用（二阶精度）Crank-Nicolson方法。我们应用一个精心定制的可逆变换来对称化矩阵，然后导出所获得的鞍点系统的最佳预条件子。这个预条件的关键组成部分是一个准确的质量矩阵近似，一个很好的近似Schur补，和一个适当的多重网格过程中应用后者的近似-这些都是使用我们的工作中构造的矩阵系统的转换。我们通过特征值的界限证明了Schur补的近似的最优性，并针对由向后欧拉离散化产生的线性系统的广泛使用的预处理器来测试我们的求解器。这些证明了我们的求解器在网格大小、正则化参数和扩散系数方面的有效性和鲁棒性。
In this article, we derive a new, fast, and robust preconditioned iterative solution strategy for the all‐at‐once solution of optimal control problems with time‐dependent PDEs as constraints, including the heat equation and the non‐steady convection–diffusion equation. After applying an optimize‐then‐discretize approach, one is faced with continuous first‐order optimality conditions consisting of a coupled system of PDEs. As opposed to most work in preconditioning the resulting discretized systems, where a (first‐order accurate) backward Euler method is used for the discretization of the time derivative, we employ a (second‐order accurate) Crank–Nicolson method in time. We apply a carefully tailored invertible transformation for symmetrizing the matrix, and then derive an optimal preconditioner for the saddle‐point system obtained. The key components of this preconditioner are an accurate mass matrix approximation, a good approximation of the Schur complement, and an appropriate multigrid process to apply this latter approximation—these are constructed using our work in transforming the matrix system. We prove the optimality of the approximation of the Schur complement through bounds on the eigenvalues, and test our solver against a widely‐used preconditioner for the linear system arising from a backward Euler discretization. These demonstrate the effectiveness and robustness of our solver with respect to mesh‐sizes, regularization parameter, and diffusion coefficient.