The ParaOpt algorithm was recently introduced as a time-parallel solver for optimal-control problems with a terminal-cost objective, and convergence results have been presented for the linear diffusive case with implicit-Euler time integrators. We reformulate ParaOpt for tracking problems and provide generalized convergence analyses for both objectives. We focus on linear diffusive equations and prove convergence bounds that are generic in the time integrators used. For large problem dimensions, ParaOpt's performance depends crucially on having a good preconditioner to solve the arising linear systems. For the case where ParaOpt's cheap, coarse-grained propagator is linear, we introduce diagonalization-based preconditioners inspired by recent advances in the ParaDiag family of methods. These preconditioners not only lead to a weakly-scalable ParaOpt version, but are themselves invertible in parallel, making maximal use of available concurrency. They have proven convergence properties in the linear diffusive case that are generic in the time discretization used, similarly to our ParaOpt results. Numerical results confirm that the iteration count of the iterative solvers used for ParaOpt's linear systems becomes constant in the limit of an increasing processor count. The paper is accompanied by a sequential MATLAB implementation.

翻译：ParaOpt算法最近被提出作为求解终端成本目标最优控制问题的时间并行求解器，并针对采用隐式欧拉时间积分器的线性扩散情形给出了收敛性结果。本文将ParaOpt重新表述为追踪问题，并为两类目标函数提供广义收敛性分析。我们聚焦于线性扩散方程，证明了时间积分器通用的收敛界。对于大规模问题维度，ParaOpt的性能关键取决于求解线性系统的优质预条件子。当ParaOpt的廉价粗粒度传播子为线性时，我们借鉴ParaDiag方法族的最新进展，引入基于对角化的预条件子。这些预条件子不仅能生成弱可扩展的ParaOpt版本，其本身即可并行求逆，从而最大化利用可用并发性。与ParaOpt结果类似，它们在线性扩散情形下具有时间离散化通用的收敛性质。数值结果证实，用于求解ParaOpt线性系统的迭代求解器的迭代次数随着处理器数目的增加而趋于常数。本文附有顺序MATLAB实现代码。