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一种面向终端成本目标函数的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问题给出了收敛性结果。本文对跟踪问题重新表述了ParaOpt算法，并为两种目标函数提供了广义收敛性分析。我们聚焦于线性扩散方程，针对所使用的时间积分器证明了通用的收敛界。对于大规模问题维度而言，ParaOpt的性能关键取决于求解线性系统时预条件子的优劣。针对ParaOpt中廉价粗粒度传播算子为线性的情形，受ParaDiag方法族最新进展启发，我们引入了基于对角化的预条件子。这些预条件子不仅使ParaOpt版本具有弱可扩展性，其本身也可并行求逆，从而最大化利用并行性。它们在线性扩散情形下具有与时间离散化格式无关的收敛性质，与我们的ParaOpt分析结论类似。数值实验表明，随着处理器数量增加，用于求解ParaOpt线性系统的迭代求解器的迭代次数趋于恒定。本文附有顺序执行MATLAB代码实现。