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cwrowley cwrowley Jan 6, 2017

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**<span style="font-size: 13px;">Koopmanism’s World Wide Quest to Rule</span>**
Shervin Bagheri and Predrag Cvitanović
(a very early draft of chapter for [[@http://ChaosBook.org|ChaosBook.org]] - any comments/edits much appreciated)
[[file:Koopmanism.pdf]]
Paper that introduces the notion of spectral expansion of the Koopman operator for model reduction and connects to Reynolds triple decomposition
<span style="background-color: #ffffff; color: #222222; font-family: Arial,sans-serif; font-size: 13px; text-align: start;">Mezić, I. (2005). Spectral properties of dynamical systems, model reduction and decompositions.</span> //<span style="background-color: #ffffff; color: #222222; font-family: Arial,sans-serif; font-size: 13px; text-align: start;">Nonlinear Dynamics</span>//<span style="background-color: #ffffff; color: #222222; font-family: Arial,sans-serif; font-size: 13px; text-align: start;">,</span> //<span style="background-color: #ffffff; color: #222222; font-family: Arial,sans-serif; font-size: 13px; text-align: start;">41</span>//<span style="background-color: #ffffff; color: #222222; font-family: Arial,sans-serif; font-size: 13px; text-align: start;">(1-3), 309-325.</span>
[[file:spectpod.pdf]]
Paper in JFM that utilizes the above idea to study jet in cross flow and show there are 2 dominant modes:
<span style="background-color: #ffffff; color: #222222; font-family: Arial,sans-serif; font-size: 13px; text-align: start;">Rowley, C.W., Mezić, I., Bagheri, S., Schlatter, P. and Henningson, D.S., 2009. Spectral analysis of nonlinear flows.</span> //<span style="background-color: #ffffff; color: #222222; font-family: Arial,sans-serif; font-size: 13px; text-align: start;">Journal of fluid mechanics</span>//<span style="background-color: #ffffff; color: #222222; font-family: Arial,sans-serif; font-size: 13px; text-align: start;">,</span> //<span style="background-color: #ffffff; color: #222222; font-family: Arial,sans-serif; font-size: 13px; text-align: start;">641</span>//<span style="background-color: #ffffff; color: #222222; font-family: Arial,sans-serif; font-size: 13px; text-align: start;">, pp.115-127.</span>
[[file:JFMKoopmanpublished.pdf]]
Paper that discusses the Koopman Mode Expansion concept and, in section .... discusses its use in analysis of the role of the mean flow :
<span style="background-color: #ffffff; color: #222222; font-family: Arial,sans-serif; font-size: 13px; text-align: start;">Mezic, I., 2013. Analysis of fluid flows via spectral properties of the Koopman operator.</span> //<span style="background-color: #ffffff; color: #222222; font-family: Arial,sans-serif; font-size: 13px; text-align: start;">Annual Review of Fluid Mechanics</span>//<span style="background-color: #ffffff; color: #222222; font-family: Arial,sans-serif; font-size: 13px; text-align: start;">,</span> //<span style="background-color: #ffffff; color: #222222; font-family: Arial,sans-serif; font-size: 13px; text-align: start;">45</span>//<span style="background-color: #ffffff; color: #222222; font-family: Arial,sans-serif; font-size: 13px; text-align: start;">, pp.357-378.</span>
<span style="background-color: #ffffff; color: #222222; font-family: Arial,sans-serif; font-size: 13px; line-height: 0px; overflow: hidden; text-align: start;">[[file:AnnualReviewsKoopman.pdf]]</span>

Paper that discusses Dynamic Mode Decomposition. See section 4.1 for the connection with the Koopman operator:
Tu, JH, Rowley, CW, Luchtenburg, DM, Brunton, SL, and Kutz, JN. On dynamic mode decomposition: theory and applications. //Journal of Computational Dynamics//, **1**(2):391-421, 2014.
[[file:2014.pdf]]
Review article on reduced-order models for fluids. See sections 4.2-4.3 for a discussion of the Koopman operator and DMD, and their use in developing reduced-order models.
Rowley, CW and Dawson, STM. Model reduction for flow analysis and control. //Annual Review of Fluid Mechanics// 49:387-417, 2017.
[[file:Rowley-ARFM-2016.pdf]]