论文标题

快速迭代求解器,以最佳控制时间依赖性PDE,并在时间内离散化。

Fast Iterative Solver for the Optimal Control of Time-Dependent PDEs with Crank-Nicolson Discretization in Time

论文作者

Leveque, Santolo, Pearson, John W.

论文摘要

在本文中,我们为最佳控制问题提供了一种新的,快速且可靠的预处理迭代解决方案策略,这些解决方案涉及时间依赖性PDE作为约束,包括热方程和非稳态对流 - 扩散方程。在应用优化的方法之后,一个人面临着由PDE耦合系统组成的连续一阶最佳条件。与大多数在预处理产生的离散系统方面的工作相反,在该系统中,使用(一阶准确)向后的Euler方法用于离散时间衍生物,我们采用了(二阶精确)曲柄 - nicolson方法。我们应用经过精心定制的可逆转换来对称矩阵,然后为获得的鞍点系统提供最佳的预处理。该预调节器的关键组成部分是准确的质量矩阵近似,良好的Schur补体近似值以及适当的多机过程以应用后一个近似值 - 这些是使用我们在转换矩阵系统中的工作来构建的。我们通过特征值的边界证明了Schur补体近似的最佳性,并测试了我们的求解器,针对由向后的Euler离散化引起的线性系统广泛使用的预处理。这些证明了求解器相对于网格尺寸,正则化参数和扩散系数的有效性和鲁棒性。

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.

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