Improved ParaDiag via low-rank updates and interpolation

TL;DR

This paper introduces two innovative low-rank update and interpolation methods for solving space-time discretized PDE matrix equations, improving efficiency and accuracy over existing ParaDiag algorithms.

Contribution

The paper presents two novel approaches leveraging low-rank updates and interpolation, enhancing the numerical solution of space-time PDE matrix equations without iterative refinement.

Findings

Methods outperform existing algorithms in several PDE cases

Avoidance of iterative refinement improves efficiency

Potential for significant performance gains demonstrated

Abstract

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.