This work is concerned with linear matrix equations that arise from the space-time discretization of time-dependent linear partial differential equations (PDEs). Such matrix equations have been considered, for example, in the context of parallel-in-time integration leading to a class of algorithms called ParaDiag. We develop and analyze two novel approaches for the numerical solution of such equations. Our first approach is based on the observation that the modification of these equations performed by ParaDiag in order to solve them in parallel has low rank. Building upon previous work on low-rank updates of matrix equations, this allows us to make use of tensorized Krylov subspace methods to account for the modification. Our second approach is based on interpolating the solution of the matrix equation from the solutions of several modifications. Both approaches avoid the use of iterative refinement needed by ParaDiag and related space-time approaches in order to attain good accuracy. In turn, our new algorithms have the potential to outperform, sometimes significantly, existing methods. This potential is demonstrated for several different types of PDEs.

这项工作涉及由时间相关线性偏微分方程 (PDE) 的时空离散化产生的线性矩阵方程。例如，在时间并行积分的背景下，已经考虑了此类矩阵方程，从而产生了一类称为 ParaDiag 的算法。我们开发并分析了两种求解此类方程的新方法。我们的第一种方法基于这样的观察：ParaDiag 为了并行求解而对这些方程进行的修改具有低秩。基于先前关于矩阵方程低秩更新的工作，这使我们能够利用张量化 Krylov 子空间方法来解释修改。我们的第二种方法是基于从几个修改的解中插值矩阵方程的解。这两种方法都避免使用 ParaDiag 和相关时空方法所需的迭代细化，以获得良好的准确性。反过来，我们的新算法有可能超越现有方法，有时甚至显着优于现有方法。几种不同类型的偏微分方程证明了这种潜力。