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Numerical approximation methods for the Koopman operator have advanced considerably in the last few years.
B. O. Koopman, “Hamiltonian systems and transformation in hilbert space,” Proceedings of the National Academy of Sciences 17
1931
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J. v. Neumann, “Zur operatorenmethode in der klassischen mechanik,” Annals of Mathematics , 587–642 (1932)
1932
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P. R. Halmos and J. von Neumann, “Operator methods in classical mechanics, ii,” Annals of Mathematics , 332–350 (1942)
1942
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A. N. Tikhonov, “On the stability of inverse problems,” in Dokl. Akad. Nauk SSSR , Vol. 39 (1943) pp. 195–198
1943
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K. Karhunen, Über lineare Methoden in der Wahrscheinlichkeitsrechnung , Vol. 37 (Universitat Helsinki, 1947)
1947
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P. R. Halmos, P. R. Halmos, P. R. Halmos, H. Mathématicien, P. R. Halmos, and H. Mathematician, Introduction to Hilbert space and the theory of spectral multiplicity (Chelsea New York, 1957)
1957
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H. Kwakernaak and R. Sivan, Linear optimal control systems , Vol. 1 (Wiley-interscience New York, 1972)
1972
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M. Loeve, “Probability theory, vol. ii,” Graduate texts in mathematics 46
1978
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M. Jolly, I. Kevrekidis, and E. Titi, “Approximate inertial manifolds for the kuramoto-sivashinsky equation: analysis and computations,” Physica D: Nonlinear Phenomena 44
1990
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T. Sauer, J. A. Yorke, and M. Casdagli, “Embedology,” Journal of statistical Physics 65
1991
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G. Berkooz, P. Holmes, and J. L. Lumley, “The proper orthogonal decomposition in the analysis of turbulent flows,” Annual review of fluid mechanics 25
1993
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Y. LeCun, Y. Bengio, et al. , “Convolutional networks for images, speech, and time series,” The handbook of brain theory and neural networks 3361
1995
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K. Engan, S. O. Aase, and J. H. Husoy, “Method of optimal directions for frame design,” in Acoustics, Speech, and Signal Processing, 1999. Proceedings., 1999 IEEE International Conference on , Vol. 5 (IEEE, 1999) pp. 2443–2446
1999
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I. Mezić and A. Banaszuk, “Comparison of systems with complex behavior,” Physica D: Nonlinear Phenomena 197
2004
Earlier work this paper cites.
A. Y. Ng, “Feature selection, l 1 vs. l 2 regularization, and rotational invariance,” in Proceedings of the twenty-first international conference on Machine learning (ACM, 2004) p. 78
2004
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I. Mezić, “Spectral properties of dynamical systems, model reduction and decompositions,” Nonlinear Dynamics 41
2005
Cited alongside, same era.
D. L. Donoho, “Compressed sensing,” IEEE Transactions on information theory 52
2006
Cited alongside, same era.
H. Lee, A. Battle, R. Raina, and A. Y. Ng, “Efficient sparse coding algorithms,” Advances in neural information processing systems 19
2007
Cited alongside, same era.
M. Aharon and M. Elad, “Sparse and redundant modeling of image content using an image-signature-dictionary,” SIAM Journal on Imaging Sciences 1
2008
Cited alongside, same era.
C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, and D. S. Henningson, “Spectral analysis of nonlinear flows,” Journal of fluid mechanics 641
2009
Cited alongside, same era.
D. Giannakis, J. Slawinska, and Z. Zhao, “Spatiotemporal feature extraction with data-driven koopman operators,” J. Mach. Learn. Res. Proceedings , 103–115 (2015)
2015
Later among the works it cites.
M. Georgescu and I. Mezić, “Building energy modeling: A systematic approach to zoning and model reduction using koopman mode analysis,” Energy and Buildings 86
2015
Later among the works it cites.
M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, “A data–driven approximation of the koopman operator: Extending dynamic mode decomposition,” Journal of Nonlinear Science 25
2015
Later among the works it cites.
Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521
2015
Later among the works it cites.
M. Budišić, R. Mohr, and I. Mezić, “Applied koopmanism,” Chaos 22
2016
Later among the works it cites.
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P. J. Schmid, “Dynamic mode decomposition of numerical and experimental data,” Journal of fluid mechanics 656
2010
Cited alongside, same era.
Y. LeCun, K. Kavukcuoglu, and C. Farabet, “Convolutional networks and applications in vision,” in Circuits and Systems (ISCAS), Proceedings of 2010 IEEE International Symposium on (IEEE, 2010) pp. 253–256
2010
Cited alongside, same era.
G. H. Golub and C. F. van Loan, Matrix computations , Vol. 3 (JHU Press, 2012)
2012
Cited alongside, same era.
P. Constantin, C. Foias, B. Nicolaenko, and R. Temam, Integral manifolds and inertial manifolds for dissipative partial differential equations , Vol. 70 (Springer Science & Business Media, 2012)
2012
Cited alongside, same era.
A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Imagenet classification with deep convolutional neural networks,” in Advances in neural information processing systems (2012) pp. 1097–1105
2012
Cited alongside, same era.
I. Mezić, “Analysis of fluid flows via spectral properties of the koopman operator,” Annual Review of Fluid Mechanics 45
2013
Cited alongside, same era.
A. Mauroy, I. Mezić, and J. Moehlis, “Isostables, isochrons, and koopman spectrum for the action–angle representation of stable fixed point dynamics,” Physica D: Nonlinear Phenomena 261
2013
Cited alongside, same era.
2016
Later among the works it cites.
A. S. Sharma, I. Mezić, and B. J. McKeon, “Correspondence between koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the navier-stokes equations,” Physical Review Fluids 1
2016
Later among the works it cites.
S. L. Brunton, J. L. Proctor, and J. N. Kutz, “Discovering governing equations from data by sparse identification of nonlinear dynamical systems,” Proceedings of the National Academy of Sciences of the United States of America 113
2016
Later among the works it cites.
S. Ruder, “An overview of gradient descent optimization algorithms,” arXiv 1609.04747
2016
Later among the works it cites.
H. Wu, F. Nüske, F. Paul, S. Klus, P. Koltai, and F. Noć, “Variational koopman models: Slow collective variables and molecular kinetics from short off-equilibrium simulations,” The Journal of Chemical Physics (2017), 10.1063/1.4979344
2017
Closest in time.
2017
Closest in time.
S. Klus, N. Peter, K. Peter, W. Hao, K. Ioannis, S. Christof, and N. Frank, “Data-driven model reduction and transfer operator approximation,” Journal of Nonlinear Science (2017), under review
2017
Closest in time.
M. Korda and I. Mezić, “On convergence of extended dynamic mode decomposition to the koopman operator,” arXiv 1703.04680v1
2017
Closest in time.
C. W. Rowley, “Data-driven methods for identifying nonlinear models of fluid flows,” http://online.kitp.ucsb.edu/online/transturb-c17/rowley/ (2017)
2017
Closest in time.