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

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...

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Bibliographic Details
Published in:BIT Vol. 63; no. 1; p. 13
Main Authors: Carrel, Benjamin, Gander, Martin J., Vandereycken, Bart
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.01.2023
Springer Nature B.V
Subjects:
Algorithms
Approximation
Computational Mathematics and Numerical Analysis
Evolution
Fields (mathematics)
Integrators
Mathematical analysis
Mathematics
Mathematics and Statistics
Numeric Computing
Runge-Kutta method
Dynamical low-rank approximation
65L05
Matrix differential equation
Parallel algorithm
65L70
65F55
Initial value problem
68W10
65L20
65F45
ISSN:0006-3835, 1572-9125, 1572-9125
Online Access:Get full text
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Summary: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.
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Communicated by Antonella Zanna Munthe-Kaas.
ISSN:0006-3835
1572-9125
1572-9125
DOI:10.1007/s10543-023-00953-3