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 non-symmetric 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：本文研究了一类由初始值抛物型偏微分方程约束的最优控制问题经二阶Crank-Nicolson离散化产生的鞍点系统的快速求解器。该鞍点系统具有不定性和病态性。与一阶Euler离散化产生的鞍点系统不同，Crank-Nicolson离散化产生的鞍点系统具有稠密且非对称的Schur补，这给快速求解器的设计带来了挑战。为解决此问题，本文对该鞍点系统应用了一种新颖的对称化技术，使得新的Schur补变为对称正定矩阵，从而著名的匹配Schur补（MSC）预条件子得以适用。然而，新的Schur补仍是稠密矩阵，且对应MSC预条件子的求逆不满足时间并行性（PinT），因此计算耗时。针对此问题，本文提出了一种针对新Schur补系统的修正MSC预条件子。该预条件子可通过时间对角化技术实现快速且时间并行的计算。理论上，本文证明了新预条件子作用下预条件矩阵的特征值可被与矩阵规模和正则化参数无关的正常数所界定。基于该谱性质，Schur补系统的预条件共轭梯度（PCG）求解器的收敛速率被证明与矩阵规模和正则化参数无关。据我们所知，这是首次针对最优控制问题Crank-Nicolson离散化产生的鞍点系统，实现问题无关收敛速率的迭代求解器。数值实验结果验证了所提预条件子的性能。