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（PDE）时空离散化产生的线性矩阵方程。此类矩阵方程已在例如并行时间积分算法的背景下被研究，进而催生了称为ParaDiag的算法类别。我们针对此类方程的数值求解提出并分析了两种新方法。第一种方法基于以下观测：ParaDiag为并行求解而对方程进行的修正具有低秩结构。基于矩阵方程低秩更新的已有工作，我们得以利用张量化的Krylov子空间方法处理该修正。第二种方法通过对多个修正方程的解进行插值来求解原矩阵方程。两种方法均避免了ParaDiag及相关时空方法为达到良好精度所需的迭代细化过程。因此，我们的新算法具有超越现有方法的潜力，在某些情况下可显著提升性能。我们通过多种不同类型PDE的数值实验验证了该潜力。