通过低秩更新与插值改进的ParaDiag
Improved ParaDiag via low-rank updates and interpolation
AI总结：
针对含时线性PDE时空离散产生的矩阵方程，提出基于低秩修正的张量化Krylov子空间法和插值求解法两种改进ParaDiag的方法，无需迭代求精即可获高精度，经多类PDE验证性能优于现有方法。
摘要：
本工作针对含时线性偏微分方程（PDE）时空离散化过程中产生的线性矩阵方程展开研究。这类矩阵方程已在例如时间并行积分的研究背景下被探讨，由此衍生出一类名为ParaDiag的算法。本文提出并分析了两种用于数值求解这类方程的新方法。第一种方法基于如下观察：ParaDiag为实现并行求解而对这些方程所做的修正具有低秩特性。基于矩阵方程低秩更新的已有研究成果，这一特性使我们能够利用张量化Krylov子空间方法来处理该修正项。第二种方法的核心是通过若干个修正后方程的解来插值得到原矩阵方程的解。两种方法都无需使用ParaDiag及相关时空方法为达到良好精度所需的迭代求精步骤。相应地，我们提出的新算法有潜力超越现有方法，有时性能提升幅度十分显著。我们通过多种不同类型的PDE验证了这一潜力。
英文摘要：
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.
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