Randomized Projection Learning Method for Dynamic Mode Decomposition

A data-driven analysis method known as dynamic mode decomposition (DMD) approximates the linear Koopman operator on a projected space. In the spirit of Johnson–Lindenstrauss lemma, we will use a random projection to estimate the DMD modes in a reduced dimensional space. In practical applications, sn...

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Autores principales:	Sudam Surasinghe, Erik M. Bollt
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	Koopman operator dynamic mode decomposition (DMD) Johnson–Lindenstrauss lemma random projection data-driven method Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/55dee2a3996c4ff5b281e86122c02833
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id	oai:doaj.org-article:55dee2a3996c4ff5b281e86122c02833
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spelling	oai:doaj.org-article:55dee2a3996c4ff5b281e86122c028332021-11-11T18:20:15ZRandomized Projection Learning Method for Dynamic Mode Decomposition10.3390/math92128032227-7390https://doaj.org/article/55dee2a3996c4ff5b281e86122c028332021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/21/2803https://doaj.org/toc/2227-7390A data-driven analysis method known as dynamic mode decomposition (DMD) approximates the linear Koopman operator on a projected space. In the spirit of Johnson–Lindenstrauss lemma, we will use a random projection to estimate the DMD modes in a reduced dimensional space. In practical applications, snapshots are in a high-dimensional observable space and the DMD operator matrix is massive. Hence, computing DMD with the full spectrum is expensive, so our main computational goal is to estimate the eigenvalue and eigenvectors of the DMD operator in a projected domain. We generalize the current algorithm to estimate a projected DMD operator. We focus on a powerful and simple random projection algorithm that will reduce the computational and storage costs. While, clearly, a random projection simplifies the algorithmic complexity of a detailed optimal projection, as we will show, the results can generally be excellent, nonetheless, and the quality could be understood through a well-developed theory of random projections. We will demonstrate that modes could be calculated for a low cost by the projected data with sufficient dimension.Sudam SurasingheErik M. BolltMDPI AGarticleKoopman operatordynamic mode decomposition (DMD)Johnson–Lindenstrauss lemmarandom projectiondata-driven methodMathematicsQA1-939ENMathematics, Vol 9, Iss 2803, p 2803 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Koopman operator dynamic mode decomposition (DMD) Johnson–Lindenstrauss lemma random projection data-driven method Mathematics QA1-939
spellingShingle	Koopman operator dynamic mode decomposition (DMD) Johnson–Lindenstrauss lemma random projection data-driven method Mathematics QA1-939 Sudam Surasinghe Erik M. Bollt Randomized Projection Learning Method for Dynamic Mode Decomposition
description	A data-driven analysis method known as dynamic mode decomposition (DMD) approximates the linear Koopman operator on a projected space. In the spirit of Johnson–Lindenstrauss lemma, we will use a random projection to estimate the DMD modes in a reduced dimensional space. In practical applications, snapshots are in a high-dimensional observable space and the DMD operator matrix is massive. Hence, computing DMD with the full spectrum is expensive, so our main computational goal is to estimate the eigenvalue and eigenvectors of the DMD operator in a projected domain. We generalize the current algorithm to estimate a projected DMD operator. We focus on a powerful and simple random projection algorithm that will reduce the computational and storage costs. While, clearly, a random projection simplifies the algorithmic complexity of a detailed optimal projection, as we will show, the results can generally be excellent, nonetheless, and the quality could be understood through a well-developed theory of random projections. We will demonstrate that modes could be calculated for a low cost by the projected data with sufficient dimension.
format	article
author	Sudam Surasinghe Erik M. Bollt
author_facet	Sudam Surasinghe Erik M. Bollt
author_sort	Sudam Surasinghe
title	Randomized Projection Learning Method for Dynamic Mode Decomposition
title_short	Randomized Projection Learning Method for Dynamic Mode Decomposition
title_full	Randomized Projection Learning Method for Dynamic Mode Decomposition
title_fullStr	Randomized Projection Learning Method for Dynamic Mode Decomposition
title_full_unstemmed	Randomized Projection Learning Method for Dynamic Mode Decomposition
title_sort	randomized projection learning method for dynamic mode decomposition
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/55dee2a3996c4ff5b281e86122c02833
work_keys_str_mv	AT sudamsurasinghe randomizedprojectionlearningmethodfordynamicmodedecomposition AT erikmbollt randomizedprojectionlearningmethodfordynamicmodedecomposition
_version_	1718431885230604288

Randomized Projection Learning Method for Dynamic Mode Decomposition

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