The ParaDiag family of algorithms solves differential equations by using preconditioners that can be inverted in parallel through diagonalization. In the context of optimal control of linear parabolic PDEs, the state-of-the-art ParaDiag method is limited to solving self-adjoint problems with a tracking objective. We propose three improvements to the ParaDiag method: the use of alpha-circulant matrices to construct an alternative preconditioner, a generalization of the algorithm for solving non-self-adjoint equations, and the formulation of an algorithm for terminal-cost objectives. We present novel analytic results about the eigenvalues of the preconditioned systems for all discussed ParaDiag algorithms in the case of self-adjoint equations, which proves the favorable properties the alpha-circulant preconditioner. We use these results to perform a theoretical parallel-scaling analysis of ParaDiag for self-adjoint problems. Numerical tests confirm our findings and suggest that the self-adjoint behavior, which is backed by theory, generalizes to the non-self-adjoint case. We provide a sequential, open-source reference solver in Matlab for all discussed algorithms.

翻译：ParaDiag算法族通过使用可并行求逆的预条件子求解微分方程，其核心机制依托对角化实现并行加速。在求解具有追踪目标的对称型线性抛物型偏微分方程最优控制问题时，现有最优解法ParaDiag方法仅适用于自伴问题。本文提出三项改进：采用α-循环矩阵构建新型预条件子，将算法推广至非自伴方程求解，以及针对终端代价目标优化算法设计。针对自伴方程情形，我们提出所有讨论的ParaDiag算法对应的预条件系统特征值解析解，该结果证实了α-循环预条件子的优越性能。基于这些发现，我们开展自伴问题的理论并行加速比分析。数值实验验证了理论结果，并表明具有理论支撑的自伴特性可推广至非自伴情形。我们提供包含所有讨论算法的开源Matlab顺序求解器作为参考实现。