论文标题
通过低级更新和插值改善了Paradiag
Improved ParaDiag via low-rank updates and interpolation
论文作者
论文摘要
这项工作涉及由时间依赖性线性偏微分方程（PDE）的时空离散化产生的线性矩阵方程。例如，在平行时间集成的背景下，已经考虑了这种矩阵方程，从而导致一类称为paradiag的算法。我们开发并分析了此类方程的数值解的两种新方法。我们的第一种方法是基于这样的观察结果，即为了并行解决这些方程式的这些方程式的修改较低。在矩阵方程的低级别更新上的先前工作的基础上，这使我们能够利用张力的Krylov子空间方法来说明修改。我们的第二种方法是基于从几种修改的解决方案中插值矩阵方程的解。两种方法都避免使用Paradiag所需的迭代精炼和相关的时空方法，以实现良好的准确性。反过来，我们的新算法有可能胜过现有方法，有时是显着的。对于几种不同类型的PDE，证明了这种潜力。
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.
