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算法族通过使用可对角化并行求逆的预条件子求解微分方程。在线性抛物型偏微分方程最优控制领域，现有ParaDiag方法仅限于求解具有跟踪目标的自伴问题。本文对ParaDiag方法提出三项改进：利用α-循环矩阵构造替代预条件子、将该算法推广至非自伴方程求解、以及提出终端成本目标函数的求解算法。针对自伴方程情形，我们推导了所有讨论的ParaDiag算法预条件系统特征值的解析结果，证明了α-循环预条件子的优良特性。基于这些结果，我们对自伴问题的ParaDiag算法进行了理论并行加速比分析。数值实验验证了理论发现，并表明有理论支撑的自伴行为可推广至非自伴情形。我们提供了所有讨论算法的顺序开源参考求解器（Matlab实现）。