In this study, the $\theta$-method is used for discretizing a class of evolutionary partial differential equations. Then, we transform the resultant all-at-once linear system and introduce a novel one-sided preconditioner, which can be fast implemented in a parallel-in-time way. By introducing an auxiliary two-sided preconditioned system, we provide theoretical insights into the relationship between the residuals of the generalized minimal residual (GMRES) method when applied to both one-sided and two-sided preconditioned systems. Moreover, we show that the condition number of the two-sided preconditioned matrix is uniformly bounded by a constant that is independent of the matrix size, which in turn implies that the convergence behavior of the GMRES method for the one-sided preconditioned system is guaranteed. Numerical experiments confirm the efficiency and robustness of the proposed preconditioning approach.

翻译：本研究采用$\theta$方法对一类演化偏微分方程进行离散化处理。随后，我们对所得的全一次性线性系统进行变换，并引入一种新颖的单侧预条件子，该预条件子能够以时间并行方式快速实现。通过引入一个辅助的双侧预条件系统，我们从理论上深入分析了广义最小残差（GMRES）方法在应用于单侧与双侧预条件系统时其残差之间的关系。此外，我们证明了双侧预条件矩阵的条件数被一个与矩阵规模无关的常数一致有界，这进而保证了GMRES方法在单侧预条件系统中的收敛行为。数值实验证实了所提预条件处理方法的效率与鲁棒性。