In this paper, a fast solver is studied for saddle point system arising from a second-order Crank-Nicolson discretization of an initial-valued parabolic PDE constrained optimal control problem, which is indefinite and ill-conditioned. Different from the saddle point system arising from the first-order Euler discretization, the saddle point system arising from Crank-Nicolson discretization has a dense and nonsymmetric Schur complement, which brings challenges to fast solver designing. To remedy this, a novel symmetrization technique is applied to the saddle point system so that the new Schur complement is symmetric definite and the well-known matching-Schur-complement (MSC) preconditioner is applicable to the new Schur complement. Nevertheless, the new Schur complement is still a dense matrix and the inversion of the corresponding MSC preconditioner is not parallel-in-time (PinT) and thus time consuming. For this concern, a modified MSC preconditioner for the new Schur complement system. Our new preconditioner can be implemented in a fast and PinT way via a temporal diagonalization technique. Theoretically, the eigenvalues of the preconditioned matrix by our new preconditioner are proven to be lower and upper bounded by positive constants independent of matrix size and the regularization parameter. With such spectrum, the preconditioned conjugate gradient (PCG) solver for the Schur complement system is proven to have a convergence rate independent of matrix size and regularization parameter. To the best of my knowledge, it is the first time to have an iterative solver with problem-independent convergence rate for the saddle point system arising from Crank-Nicolson discretization of the optimal control problem. Numerical results are reported to show that the performance of the proposed preconditioner.

翻译：本文针对初始值抛物型偏微分方程约束最优控制问题的二阶Crank-Nicolson离散化所导出的鞍点系统（该系统为不定且病态）研究了一种快速求解器。与一阶欧拉离散化导出的鞍点系统不同，Crank-Nicolson离散化产生的鞍点系统具有稠密且非对称的舒尔补，给快速求解器设计带来挑战。为解决此问题，本文对鞍点系统应用了一种新型对称化技术，使得新舒尔补为对称正定，从而可应用经典的匹配舒尔补（MSC）预条件子。然而新舒尔补仍为稠密矩阵，对应MSC预条件子的求逆无法实现时域并行（PinT），因而计算耗时。针对这一问题，本文提出了一种改进的MSC预条件子用于新舒尔补系统。该预条件子可通过时域对角化技术以快速且时域并行的方式实现。理论证明，经新预条件子预处理后矩阵的特征值上下界均为与矩阵规模和正则化参数无关的正常数。基于该谱性质，舒尔补系统的预条件共轭梯度（PCG）求解器的收敛速度与矩阵规模和正则化参数无关。据我们所知，这是首次为Crank-Nicolson离散化最优控制问题导出的鞍点系统建立具有问题无关收敛率的迭代求解器。数值实验结果验证了所提预条件子的性能。