In this article, we derive fast and robust parallel-in-time preconditioned iterative methods for the all-at-once linear systems arising upon discretization of time-dependent PDEs. The discretization we employ is based on a Runge--Kutta method in time, for which the development of parallel solvers is an emerging research area in the literature of numerical methods for time-dependent PDEs. By making use of classical theory of block matrices, one is able to derive a preconditioner for the systems considered. The block structure of the preconditioner allows for parallelism in the time variable, as long as one is able to provide an optimal solver for the system of the stages of the method. We thus propose a preconditioner for the latter system based on a singular value decomposition (SVD) of the (real) Runge--Kutta matrix $A_{\mathrm{RK}} = U \Sigma V^\top$. Supposing $A_{\mathrm{RK}}$ is invertible, we prove that the spectrum of the system for the stages preconditioned by our SVD-based preconditioner is contained within the right-half of the unit circle, under suitable assumptions on the matrix $U^\top V$ (the assumptions are well posed due to the polar decomposition of $A_{\mathrm{RK}}$). We show the numerical efficiency of our SVD-based preconditioner by solving the system of the stages arising from the discretization of the heat equation and the Stokes equations, with sequential time-stepping. Finally, we provide numerical results of the all-at-once approach for both problems, showing the speed-up achieved on a parallel architecture.

翻译：本文针对时间依赖偏微分方程离散化过程中产生的全时段线性系统，推导了快速且鲁棒的并行时间预条件迭代方法。所采用的离散化基于时间方向的龙格-库塔方法，该方法对应的并行求解器开发已成为时变偏微分方程数值方法领域的新兴研究方向。借助经典的块矩阵理论，可为所考虑的系统推导出预条件子。该预条件子的块结构支持时间维度的并行性，前提是可为目标方法阶段方程组提供最优求解器。基于此，我们利用（实）龙格-库塔矩阵 $A_{\mathrm{RK}} = U \Sigma V^\top$ 的奇异值分解（SVD）提出了一种预条件子。假设 $A_{\mathrm{RK}}$ 可逆，在关于矩阵 $U^\top V$ 的适当假设条件下（由于 $A_{\mathrm{RK}}$ 的极分解，该假设是适定的），我们证明了经SVD预条件子处理后的阶段方程组谱位于单位圆右半平面。通过求解热方程和斯托克斯方程离散化产生的阶段方程组（采用顺序时间推进），我们展示了基于SVD预条件子的数值效率。最后，给出两个问题的全时段方法数值结果，展示了在并行架构上实现的加速比。