Low-rank Parareal: a low-rank parallel-in-time integrator

TL;DR

This paper introduces low-rank Parareal, a novel parallel-in-time integrator leveraging low-rank approximations and dynamical low-rank approximation techniques to efficiently solve evolution problems.

Contribution

It presents a new low-rank Parareal algorithm that combines parallel-in-time integration with low-rank approximation methods for evolution problems.

Findings

Analyzed on affine linear problems.

Numerical illustrations demonstrate effectiveness.

Reduces computational cost for low-rank evolution problems.

Abstract

In this work, the Parareal algorithm is applied to evolution problems that admit good low-rank approximations and for which the dynamical low-rank approximation (DLRA) can be used as time stepper. Many discrete integrators for DLRA have recently been proposed, based on splitting the projected vector field or by applying projected Runge--Kutta methods. The cost and accuracy of these methods are mostly governed by the rank chosen for the approximation. These properties are used in a new method, called low-rank Parareal, in order to obtain a time-parallel DLRA solver for evolution problems. The algorithm is analyzed on affine linear problems and the results are illustrated numerically.