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A preconditioned Multiple Shooting Shadowing algorithm for the sensitivity analysis of chaotic systems

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Title: A preconditioned Multiple Shooting Shadowing algorithm for the sensitivity analysis of chaotic systems
Authors: Shawki, K
Papadakis, G
Item Type: Journal Article
Abstract: We propose a preconditioner that can accelerate the rate of convergence of the Multiple Shooting Shadowing (MSS) method [1]. This recently proposed method can be used to compute derivatives of time-averaged objectives (also known as sensitivities) to system parameter(s) for chaotic systems. We propose a block diagonal preconditioner, which is based on a partial singular value decomposition of the MSS constraint matrix. The preconditioner can be computed using matrix-vector products only (i.e. it is matrix-free) and is fully parallelised in the time domain. Two chaotic systems are considered, the Lorenz system and the 1D Kuramoto Sivashinsky equation. Combination of the preconditioner with a regularisation method leads to tight bracketing of the eigenvalues to a narrow range. This combination results in a significant reduction in the number of iterations, and renders the convergence rate almost independent of the number of degrees of freedom of the system, and the length of the trajectory that is used to compute the time-averaged objective. This can potentially allow the method to be used for large chaotic systems (such as turbulent flows) and optimal control applications. The singular value decomposition of the constraint matrix can also be used to quantify the effect of the system condition on the accuracy of the sensitivities. In fact, neglecting the singular modes affected by noise, we recover the correct values of sensitivity that match closely with those obtained with finite differences for the Kuramoto Sivashinsky equation in the light turbulent regime. We notice a similar improvement for the Lorenz system as well.
Issue Date: 1-Dec-2019
Date of Acceptance: 27-Jul-2019
URI: http://hdl.handle.net/10044/1/72202
DOI: 10.1016/j.jcp.2019.108861
ISSN: 0021-9991
Publisher: Elsevier
Start Page: 1
End Page: 19
Journal / Book Title: Journal of Computational Physics
Volume: 398
Issue: 1
Copyright Statement: © 2019 Elsevier Ltd. All rights reserved. This manuscript is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Licence http://creativecommons.org/licenses/by-nc-nd/4.0/
Keywords: Science & Technology
Technology
Physical Sciences
Computer Science, Interdisciplinary Applications
Physics, Mathematical
Computer Science
Physics
Sensitivity analysis
Chaotic systems
Least Squares Shadowing
Preconditioning
DYNAMICAL-SYSTEMS
EFFICIENT METHOD
L-CURVE
CONTINUATION
CLIMATE
01 Mathematical Sciences
02 Physical Sciences
09 Engineering
Applied Mathematics
Publication Status: Published
Article Number: 108861
Online Publication Date: 2019-08-02
Appears in Collections:Aeronautics
Faculty of Engineering