Improved ParaDiag via low-rank updates and interpolation

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 approaches have the potential to outperform, sometimes significantly, existing approaches. This potential is demonstrated for several different types of PDEs.

On Generalized Preconditioners for Time-Parallel Parabolic Optimal Control

“…Adapting these methods to optimization ParaDiag could substantially improve performance. Lastly, a very interesting recent result [18] in the domain of ivp ParaDiag suggests using alpha-circulant approximations, but not as preconditioners. Instead, it is noted that in the ivp situation, the exact system matrix is P (α) with α = 0 and its inversion is seen as an interpolation problem, with as data points several inversions with α j = 0.…”

“…However, for the tracking preconditioner (2.10), P (0) = A does hold. As this text makes α = 1 feasible for tracking, we can use different α j with magnitude 1 as data points and [18]'s technique could now apply to tracking-type optimal-control ParaDiag.…”