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.

翻译：本文关注由含时线性偏微分方程时空离散化产生的线性矩阵方程。此类矩阵方程已在并行时间积分框架中被广泛研究，并衍生出名为ParaDiag的算法体系。我们提出并分析了两种求解该类方程的新方法。第一种方法基于以下观察：ParaDiag为实现并行求解而对方程进行的修改具有低秩结构。基于矩阵方程低秩更新的既有研究，我们利用张量化Krylov子空间方法处理该修改。第二种方法通过组合多种修改问题的解来插值原矩阵方程的解。两种方法均无需采用ParaDiag及其相关时空方法为保证精度所需的迭代校正步骤。因此，我们的新算法在性能上可能显著超越现有方法——这一潜力在多种不同类型的偏微分方程数值算例中得到验证。