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Recent years have seen rapid advances in the data-driven analysis of dynamical systems based on Koopman operator theory and related approaches.
Hamiltonian systems and transformation in Hilbert space
B. O. Koopman · 1931
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Simultaneous state-time approximation of the chemical master equation using tensor product formats
S. Dolgov and B. Khoromskij · 1942
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An ergodic theorem for operators satisfying norm conditions
R. V. Chacon · 1962
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The extension of factor analysis to three-dimensional matrices
L. R. Tucker · 1964
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A generalized solution of the orthogonal procrustes problem
P. H. Schönemann · 1966
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Analysis of individual differences in multidimensional scaling via an N-way generalization of ’Eckart-Young’ decomposition
J. D. Carroll and J. J. Chang · 1970
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Applications of negative dimensional tensors
R. Penrose · 1971
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Rigorous results on valence-bond ground states in antiferromagnets
I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki · 1987
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AMUSE: a new blind identification algorithm
L. Tong, V. C. Soon, Y. F. Huang, and R. Liu · 1990
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Matrix perturbation theory
G. W. Stewart and J. Sun · 1990
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Thermodynamic limit of density matrix renormalization
S. Östlund and S. Rommer · 1995
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On the approximation of complicated dynamical behavior
M. Dellnitz and O. Junge · 1999
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A direct approach to conformational dynamics based on hybrid Monte Carlo
C. Schütte, A. Fischer, W. Huisinga, and P. Deuflhard · 1999
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The multiconfiguration time-dependent Hartree (MCTDH) method: a highly efficient algorithm for propagating wavepackets
M. H. Beck, A. Jäckle, G. A. Worth, and H. D. Meyer · 2000
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The maximal-volume concept in approximation by low-rank matrices
S. A. Goreinov and E. E. Tyrtyshnikov · 2001
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Principal angles between subspaces in an A-based scalar product: algorithms and perturbation estimates
A. V. Knyazev and M. E. Argentati · 2002
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Canonical correlation analysis: An overview with application to learning methods
D. R. Hardoon, S. Szedmak, and J. Shawe-Taylor · 2004
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Robust Perron cluster analysis in conformation dynamics
P Deuflhard and M. Weber · 2004
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Spectral properties of dynamical systems, model reduction and decompositions
I. Mezić · 2005
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A new scheme for the tensor representation
W. Hackbusch and S. Kühn · 2009
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Breaking the curse of dimensionality, or how to use SVD in many dimensions
I. Oseledets and E. Tyrtyshnikov · 2009
Cited alongside, same era.
TT-cross approximation for multidimensional arrays
I. Oseledets and E. Tyrtyshnikov · 2009
Cited alongside, same era.
Almost-invariant sets and invariant manifolds — Connecting probabilistic and geometric descriptions of coherent structures in flows
G. Froyland and K. Padberg · 2009
Cited alongside, same era.
Transport in time-dependent dynamical systems: Finite-time coherent sets
G. Froyland, N. Santitissadeekorn, and A. Monahan · 2010
Cited alongside, same era.
How to find a good submatrix
S. A. Goreinov, I. V. Oseledets, D. V. Savostyanov, E. E. Tyrtyshnikov, and N. L. Zamarashkin · 2010
Cited alongside, same era.
On the approximation quality of Markov state models
M. Sarich, F. Noé, and C. Schütte · 2010
On the numerical approximation of the Perron–Frobenius and Koopman operator
S. Klus, P. Koltai, and C. Schütte · 2016
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Solving the Master Equation Without Kinetic Monte Carlo
P. Gelß, S. Matera, and C. Schütte · 2016
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Tensor computation: A new framework for high-dimensional problems in EDA
Z. Zhang, K. Batselier, H. Liu, L. Daniel, and N. Wong · 2016
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Towards tensor-based methods for the numerical approximation of the Perron–Frobenius and Koopman operator
S. Klus and C. Schütte · 2016
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Variational tensor approach for approximating the rare-event kinetics of macromolecular systems
F. Nüske, R. Schneider, F. Vitalini, and F. Noé · 2016
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Understanding the geometry of transport: Diffusion maps for Lagrangian trajectory data unravel coherent sets
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Cited alongside, same era.
Tensor-train decomposition
I. Oseledets · 2011
Cited alongside, same era.
How fast-folding proteins fold
K. Lindorff-Larsen, S. Piana, R. O. Dror, and D. E. Shaw · 2011
Cited alongside, same era.
Markov models of molecular kinetics: Generation and validation
J.-H. Prinz, H. Wu, M. Sarich, B. Keller, M. Senne, M. Held, J. D. Chodera, C. Schütte, and F. Noé · 2011
Cited alongside, same era.
An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems
G. Froyland · 2013
Cited alongside, same era.
A variational approach to modeling slow processes in stochastic dynamical systems
F. Noé and F. Nüske · 2013
Cited alongside, same era.
Computation of extreme eigenvalues in higher dimensions using block tensor train format
S. V. Dolgov, B. N. Khoromskij, I. V. Oseledets, and D. V. Savostyanov · 2013
Cited alongside, same era.
R. Banisch and P. Koltai · 2017
Later among the works it cites.
Nearest-neighbor interaction systems in the tensor-train format
P. Gelß, S. Klus, S. Matera, and C. Schütte · 2017
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A data-driven perspective on the hierarchical assembly of molecular structures
L. Boninsegna, R. Banisch, and C. Clementi · 2017
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Data-Driven Model Reduction and Transfer Operator Approximation
S. Klus, F. Nüske, P. Koltai, H. Wu, I. Kevrekidis, C. Schütte, and F. Noé · 2018
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Deep learning for universal linear embeddings of nonlinear dynamics
B. Lusch, J. N. Kutz, and S. L. Brunton · 2018
Later among the works it cites.
VAMPnets for deep learning of molecular kinetics
A. Mardt, L. Pasquali, H. Wu, and F. Noé · 2018
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Tensor-based dynamic mode decomposition
S. Klus, P. Gelß, S. Peitz, and C. Schütte · 2018
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Rapid calculation of molecular kinetics using compressed sensing
F. Litzinger, L. Boninsegna, H. Wu, F. Nüske, R. Patel, R. Baraniuk, F. Noé, and C. Clementi · 2018
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On convergence of extended dynamic mode decomposition to the Koopman operator
M. Korda and I. Mezić · 2018
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Kernel methods for detecting coherent structures in dynamical data
S. Klus, B. E. Husic, M. Mollenhauer, and F. Noé · 2019
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Multidimensional Approximation of Nonlinear Dynamical Systems
P. Gelß, S. Klus, J. Eisert, and C. Schütte · 2019
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Variational approach for learning Markov processes from time series data
H. Wu and F. Noé · 2020
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Kernel-based approximation of the Koopman generator and Schrödinger operator
S. Klus, F. Nüske, and B. Hamzi · 2020
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Eigendecompositions of transfer operators in reproducing kernel Hilbert spaces
S. Klus, I. Schuster, and K. Muandet · 2020
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