一种基于傅里叶感知投影的周期并行实时算法求解时间周期问题
A Fourier-Aware Projection-Based Periodic Parareal Method for Time-Periodic Problems
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中文总结 AI 辅助
研究针对时间周期问题，提出傅里叶感知投影空间构造及新校正方案，用于加速基于投影的周期并行实时算法收敛。经收敛性分析与实验验证，该方法在解决线性和非线性问题时，比克里洛夫增强算法外层迭代更少，效果更好。
中文摘要 AI 辅助
当期望的解是周期稳态而非瞬态轨迹时会出现时间周期问题。具有周期粗问题的周期并行实时算法（PP-PC）是解决此类问题的一种保持周期性的并行实时方法。基于投影的校正可加速并行实时算法和PP-PC的收敛。本文提出了一种傅里叶感知投影空间构造和新的校正方案，以进一步加速基于投影的PP-PC的收敛。我们针对一般非线性时间周期问题，对基于投影的PP-PC与基于差异的校正方案进行了收敛性分析。对于任意正交投影，我们推导了由未解决误差和显式非线性贡献控制的局部单步收敛估计。时间傅里叶分解通过尾漏量来界定未解决误差，当选择主导误差模式并由投影空间捕获其系数时，尾漏量较小。对于线性问题，非线性贡献消失，在较弱假设下产生全局有效的单步尾漏收敛估计。线性和非线性问题的实验表明，傅里叶感知PP-PC比克里洛夫增强PP-PC需要更少的外层迭代。对于线性问题，误差跟踪尾漏界。对于非线性问题，实验量化了局部单步估计中的未解决误差和显式非线性贡献，并表明评估的尾漏估计遵循观察到的衰减。
英文摘要
Time-periodic problems arise when the desired solution is a periodic steady state rather than a transient trajectory. The periodic parareal algorithm with a periodic coarse problem (PP-PC) is a periodicity-preserving parallel-in-time approach for such problems. Projection-based correction can accelerate convergence of both parareal and PP-PC. In this paper, we propose a Fourier-aware construction of projection spaces and a new correction scheme to further accelerate the convergence of projection-based PP-PC. We develop a convergence analysis of projection-based PP-PC with the discrepancy-based correction scheme for general nonlinear time-periodic problems. For an arbitrary orthogonal projection, we derive a local one-step convergence estimate controlled by the unresolved error and explicit nonlinear contributions. A temporal Fourier decomposition bounds the unresolved error by a tail-leak quantity, which is small when dominant error modes are selected and their coefficients are captured by the projection space. For linear problems, the nonlinear contributions vanish, yielding a globally valid one-step tail-leak convergence estimate under weaker assumptions. Experiments on linear and nonlinear problems show that Fourier-aware PP-PC requires fewer outer iterations than Krylov-enhanced PP-PC. For the linear problems, the errors track the tail-leak bound. For the nonlinear problems, the experiments quantify the unresolved-error and explicit nonlinear contributions in the local one-step estimate and show that the evaluated tail-leak estimate follows the observed decay.
