2016, 3(1): 51-79. doi: 10.3934/jcd.2016003

On the numerical approximation of the Perron-Frobenius and Koopman operator

1. 

Department of Mathematics and Computer Science, Freie Universität Berlin, Germany, Germany

2. 

Freie Universität Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin

Received  December 2015 Revised  July 2016 Published  September 2016

Information about the behavior of dynamical systems can often be obtained by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with a dynamical system. Examples of such operators are the Perron-Frobenius and the Koopman operator. In this paper, we will review di erent methods that have been developed over the last decades to compute nite-dimensional approximations of these in nite-dimensional operators - in particular Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and highlight the similarities and di erences between these approaches. The results will be illustrated using simple stochastic di erential equations and molecular dynamics examples.
Citation: Stefan Klus, Péter Koltai, Christof Schütte. On the numerical approximation of the Perron-Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (1) : 51-79. doi: 10.3934/jcd.2016003
References:
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J. R. Baxter and J. S. Rosenthal, Rates of convergence for everywhere-positive Markov chains,, Statistics & Probability Letters, 22 (1995), 333. doi: 10.1016/0167-7152(94)00085-M.

[2]

A. Bittracher, P. Koltai and O. Junge, Pseudogenerators of spatial transfer operators,, SIAM Journal on Applied Dynamical Systems, 14 (2015), 1478. doi: 10.1137/14099872X.

[3]

E. M. Bollt and N. Santitissadeekorn, Applied and Computational Measurable Dynamics,, Society for Industrial and Applied Mathematics, (2013). doi: 10.1137/1.9781611972641.

[4]

C. J. Bose and R. Murray, The exact rate of approximation in Ulam's method,, Discrete and Continuous Dynamical Systems, 7 (2001), 219.

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C. J. Bose and R. Murray, Dynamical conditions for convergence of a maximum entropy method for Frobenius-Perron operator equations,, Applied Mathematics and Computation, 182 (2006), 210. doi: 10.1016/j.amc.2006.01.089.

[6]

C. J. Bose and R. Murray, Minimum 'energy' approximations of invariant measures for nonsingular transformations,, Discrete and Continuous Dynamical Systems, 14 (2006), 597.

[7]

C. J. Bose and R. Murray, Duality and the computation of approximate invariant densities for nonsingular transformations,, SIAM Journal on Optimization, 18 (2007), 691. doi: 10.1137/060658163.

[8]

A. Boyarsky and P. Gora, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension,, Probability and its Applications. Birkhäuser Boston, (1997). doi: 10.1007/978-1-4612-2024-4.

[9]

J. P. Boyd, Chebyshev and Fourier Spectral Methods,, 2nd edition, (2001).

[10]

M. Budišić, R. Mohr and I. Mezić, Applied Koopmanism,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 ().

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H.-J. Bungartz and M. Griebel, Sparse grids,, Acta Numerica, 13 (2004), 147. doi: 10.1017/S0962492904000182.

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D. A. Case, J. T. Berryman, R. M. Betz, D. S. Cerutti, T. E. Cheatham, T. A. Darden, R. E. Duke, T. J. Giese, H. Gohlke, A. W. Goetz, N. Homeyer, S. Izadi, P. Janowski, J. Kaus, A. Kovalenko, T. S. Lee, S. LeGrand, P. Li, T. Luchko, R. Luo, B. Madej, K. M. Merz, G. Monard, P. Needham, H. Nguyen, H. T. Nguyen, I. Omelyan, A. Onufriev, D. R. Roe, A. Roitberg, R. Salomon-Ferrer, C. L. Simmerling, W. Smith, J. Swails, R. C. Walker, J. Wang, R. M. Wolf, X. Wu, D. M. York and P. A. Kollman, AMBER 2015,, University of California, (2015).

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M. D. Chekroun, J. D. Neelin, D. Kondrashov, J. C. McWilliams and M. Ghil, Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonances,, Proceedings of the National Academy of Sciences, 111 (2014), 1684. doi: 10.1073/pnas.1321816111.

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M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM Journal on Numerical Analysis, 36 (1999), 491. doi: 10.1137/S0036142996313002.

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[20]

J. Ding and A. Zhou, Finite approximations of Frobenius-Perron operators. A solution of Ulam's conjucture on multi-dimensional transformations,, Physica D, 92 (1996), 61. doi: 10.1016/0167-2789(95)00292-8.

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G. Froyland, C. González-Tokman and T. M. Watson, Optimal mixing enhancement by local perturbation,, Preprint., ().

[23]

G. Froyland, G. Gottwald and A. Hammerlindl, A computational method to extract macroscopic variables and their dynamics in multiscale systems,, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1816. doi: 10.1137/130943637.

[24]

G. Froyland, R. M. Stuart and E. van Sebille, How well-connected is the surface of the global ocean?,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 24 (2014). doi: 10.1063/1.4892530.

[25]

G. Froyland, Approximating physical invariant measures of mixing dynamical systems,, Nonlinear Analysis, 32 (1998), 831. doi: 10.1016/S0362-546X(97)00527-0.

[26]

G. Froyland and O. Junge, On fast computation of finite-time coherent sets using radial basis functions,, Chaos, 25 (2015). doi: 10.1063/1.4927640.

[27]

G. Froyland, O. Junge and P. Koltai, Estimating long term behavior of flows without trajectory integration: The infinitesimal generator approach,, SIAM Journal on Numerical Analysis, 51 (2013), 223. doi: 10.1137/110819986.

[28]

P. R. Halmos, Lectures on Ergodic Theory, vol. 142,, American Mathematical Soc., (1956).

[29]

E. Hopf, The general temporally discrete Markoff process,, Journal of Rational Mechanics and Analysis, 3 (1954), 13.

[30]

P. Huber, Dünngitter-Spektralmethoden zur Approximation des Frobenius-Perron-Operators,, Diploma thesis (in German), (2009).

[31]

M. R. Jovanović, P. J. Schmid and J. W. Nichols, Sparsity-promoting dynamic mode decomposition,, Physics of Fluids, 26 ().

[32]

O. Junge and P. Koltai, Discretization of the Frobenius-Perron operator using a sparse Haar tensor basis: The Sparse Ulam method,, SIAM Journal on Numerical Analysis, 47 (2009), 3464. doi: 10.1137/080716864.

[33]

P. Koltai, Efficient Approximation Methods for the Global Long-Term Behavior of Dynamical Systems - Theory, Algorithms and Examples,, PhD thesis, (2010).

[34]

B. O. Koopman, Hamiltonian systems and transformation in Hilbert space,, Proceedings of the National Academy of Sciences of the United States of America, 17 (1931), 315. doi: 10.1073/pnas.17.5.315.

[35]

U. Krengel, Ergodic Theorems, vol. 6 of de Gruyter Studies in Mathematics,, Walter de Gruyter & Co., (1985). doi: 10.1515/9783110844641.

[36]

A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, vol. 97 of Applied Mathematical Sciences,, 2nd edition, (1994). doi: 10.1007/978-1-4612-4286-4.

[37]

T.-Y. Li, Finite approximation for the Frobenius-Perron operator. A solution to Ulam's conjecture,, Journal of Approximation Theory, 17 (1976), 177. doi: 10.1016/0021-9045(76)90037-X.

[38]

J. C. Mattingly and A. M. Stuart, Geometric ergodicity of some hypo-elliptic diffusions for particle motions,, Markov Process. Related Fields, 8 (2002), 199.

[39]

S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability,, Springer Science & Business Media, (2012).

[40]

R. Murray, Discrete Approximation of Invariant Densities,, PhD thesis, (1997).

[41]

R. Murray, Optimal partition choice for invariant measure approximation for one-dimensional maps,, Nonlinearity, 17 (2004), 1623. doi: 10.1088/0951-7715/17/5/004.

[42]

F. Noé and F. Nüske, A variational approach to modeling slow processes in stochastic dynamical systems,, Multiscale Modeling & Simulation, 11 (2013), 635. doi: 10.1137/110858616.

[43]

F. Nüske, B. G. Keller, G. Pérez-Hernández, A. S. J. S. Mey and F. Noé, Variational approach to molecular kinetics,, Journal of Chemical Theory and Computation, 10 (2014), 1739.

[44]

F. Nüske, R. Schneider, F. Vitalini and F. Noé, Variational tensor approach for approximating the rare-event kinetics of macromolecular systems,, The Journal of Chemical Physics, 144 ().

[45]

S. Ober-Blöbaum and K. Padberg-Gehle, Multiobjective optimal control of fluid mixing,, PAMM, 15 (2015), 639.

[46]

D. Ornstein, Bernoulli shifts with the same entropy are isomorphic,, Advances in Mathematics, 4 (1970), 337. doi: 10.1016/0001-8708(70)90029-0.

[47]

R. Preis, M. Dellnitz, M. Hessel, C. Schütte and E. Meerbach, Dominant Paths Between Almost Invariant Sets of Dynamical Systems,, DFG Schwerpunktprogramm 1095, (1095).

[48]

P. Schmid and J. Sesterhenn, Dynamic mode decomposition of numerical and experimental data,, in 61st Annual Meeting of the APS Division of Fluid Dynamics, (2008).

[49]

Schrödinger, LLC, The PyMOL molecular graphics system,, Version 1.7.4, (2014).

[50]

C. Schütte, Conformational Dynamics: Modelling, Theory, Algorithm, and Application to Biomolecules, 1999,, Habilitation Thesis., ().

[51]

C. Schütte and M. Sarich, Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches,, no. 24 in Courant Lecture Notes, (2013).

[52]

Y. G. Sinai, On the notion of entropy of dynamical systems,, in Doklady Akademii Nauk, 124 (1959), 768.

[53]

A. Tantet, V. Lucarini, F. Lunkeit and H. A. Dijkstra, Crisis of the chaotic attractor of a climate model: A transfer operator approach,, Preprint, ().

[54]

A. Tantet, F. R. van der Burgt and H. A. Dijkstra, An early warning indicator for atmospheric blocking events using transfer operators,, Chaos, 25 (2015). doi: 10.1063/1.4908174.

[55]

J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton and J. N. Kutz, On dynamic mode decomposition: Theory and applications,, ArXiv e-prints., ().

[56]

S. M. Ulam, A Collection of Mathematical Problems,, Interscience Publisher NY, (1960).

[57]

U. Vaidya, P. G. Mehta and U. V. Shanbhag, Nonlinear stabilization via control Lyapunov measure,, IEEE Transactions on Automatic Control, 55 (2010), 1314. doi: 10.1109/TAC.2010.2042226.

[58]

M. O. Williams, I. G. Kevrekidis and C. W. Rowley, A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition,, J. Nonlinear Sci., 25 (2015), 1307. doi: 10.1007/s00332-015-9258-5.

[59]

M. O. Williams, C. W. Rowley and I. G. Kevrekidis, A kernel-based approach to data-driven Koopman spectral analysis,, J. Comput. Dyn., 2 (2015), 247. doi: 10.3934/jcd.2015005.

[60]

M. O. Williams, I. I. Rypina and C. W. Rowley, Identifying finite-time coherent sets from limited quantities of Lagrangian data,, Chaos, 25 (2015). doi: 10.1063/1.4927424.

show all references

References:
[1]

J. R. Baxter and J. S. Rosenthal, Rates of convergence for everywhere-positive Markov chains,, Statistics & Probability Letters, 22 (1995), 333. doi: 10.1016/0167-7152(94)00085-M.

[2]

A. Bittracher, P. Koltai and O. Junge, Pseudogenerators of spatial transfer operators,, SIAM Journal on Applied Dynamical Systems, 14 (2015), 1478. doi: 10.1137/14099872X.

[3]

E. M. Bollt and N. Santitissadeekorn, Applied and Computational Measurable Dynamics,, Society for Industrial and Applied Mathematics, (2013). doi: 10.1137/1.9781611972641.

[4]

C. J. Bose and R. Murray, The exact rate of approximation in Ulam's method,, Discrete and Continuous Dynamical Systems, 7 (2001), 219.

[5]

C. J. Bose and R. Murray, Dynamical conditions for convergence of a maximum entropy method for Frobenius-Perron operator equations,, Applied Mathematics and Computation, 182 (2006), 210. doi: 10.1016/j.amc.2006.01.089.

[6]

C. J. Bose and R. Murray, Minimum 'energy' approximations of invariant measures for nonsingular transformations,, Discrete and Continuous Dynamical Systems, 14 (2006), 597.

[7]

C. J. Bose and R. Murray, Duality and the computation of approximate invariant densities for nonsingular transformations,, SIAM Journal on Optimization, 18 (2007), 691. doi: 10.1137/060658163.

[8]

A. Boyarsky and P. Gora, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension,, Probability and its Applications. Birkhäuser Boston, (1997). doi: 10.1007/978-1-4612-2024-4.

[9]

J. P. Boyd, Chebyshev and Fourier Spectral Methods,, 2nd edition, (2001).

[10]

M. Budišić, R. Mohr and I. Mezić, Applied Koopmanism,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 ().

[11]

H.-J. Bungartz and M. Griebel, Sparse grids,, Acta Numerica, 13 (2004), 147. doi: 10.1017/S0962492904000182.

[12]

D. A. Case, J. T. Berryman, R. M. Betz, D. S. Cerutti, T. E. Cheatham, T. A. Darden, R. E. Duke, T. J. Giese, H. Gohlke, A. W. Goetz, N. Homeyer, S. Izadi, P. Janowski, J. Kaus, A. Kovalenko, T. S. Lee, S. LeGrand, P. Li, T. Luchko, R. Luo, B. Madej, K. M. Merz, G. Monard, P. Needham, H. Nguyen, H. T. Nguyen, I. Omelyan, A. Onufriev, D. R. Roe, A. Roitberg, R. Salomon-Ferrer, C. L. Simmerling, W. Smith, J. Swails, R. C. Walker, J. Wang, R. M. Wolf, X. Wu, D. M. York and P. A. Kollman, AMBER 2015,, University of California, (2015).

[13]

M. D. Chekroun, J. D. Neelin, D. Kondrashov, J. C. McWilliams and M. Ghil, Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonances,, Proceedings of the National Academy of Sciences, 111 (2014), 1684. doi: 10.1073/pnas.1321816111.

[14]

G. Chen and T. Ueta (eds.), Chaos in Circuits and Systems,, World Scientific Series on Nonlinear Science, (2002). doi: 10.1142/9789812705303.

[15]

M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - Set oriented numerical methods for dynamical systems,, in Ergodic theory, (2001), 145.

[16]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM Journal on Numerical Analysis, 36 (1999), 491. doi: 10.1137/S0036142996313002.

[17]

J. Ding, A maximum entropy method for solving Frobenius-Perron operator equations,, Applied Mathematics and Computation, 93 (1998), 155. doi: 10.1016/S0096-3003(97)10061-3.

[18]

J. Ding, Q. Du and T.-Y. Li, High order approximation of the Frobenius-Perron operator,, Applied Mathematics and Computation, 53 (1993), 151. doi: 10.1016/0096-3003(93)90099-Z.

[19]

J. Ding and T.-Y. Li, Markov finite approximation of the Frobenius-Perron operator,, Nonlinear Analysis: Theory, 17 (1991), 759. doi: 10.1016/0362-546X(91)90211-I.

[20]

J. Ding and A. Zhou, Finite approximations of Frobenius-Perron operators. A solution of Ulam's conjucture on multi-dimensional transformations,, Physica D, 92 (1996), 61. doi: 10.1016/0167-2789(95)00292-8.

[21]

H. Federer, Geometric Measure Theory,, Springer New York, (1969).

[22]

G. Froyland, C. González-Tokman and T. M. Watson, Optimal mixing enhancement by local perturbation,, Preprint., ().

[23]

G. Froyland, G. Gottwald and A. Hammerlindl, A computational method to extract macroscopic variables and their dynamics in multiscale systems,, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1816. doi: 10.1137/130943637.

[24]

G. Froyland, R. M. Stuart and E. van Sebille, How well-connected is the surface of the global ocean?,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 24 (2014). doi: 10.1063/1.4892530.

[25]

G. Froyland, Approximating physical invariant measures of mixing dynamical systems,, Nonlinear Analysis, 32 (1998), 831. doi: 10.1016/S0362-546X(97)00527-0.

[26]

G. Froyland and O. Junge, On fast computation of finite-time coherent sets using radial basis functions,, Chaos, 25 (2015). doi: 10.1063/1.4927640.

[27]

G. Froyland, O. Junge and P. Koltai, Estimating long term behavior of flows without trajectory integration: The infinitesimal generator approach,, SIAM Journal on Numerical Analysis, 51 (2013), 223. doi: 10.1137/110819986.

[28]

P. R. Halmos, Lectures on Ergodic Theory, vol. 142,, American Mathematical Soc., (1956).

[29]

E. Hopf, The general temporally discrete Markoff process,, Journal of Rational Mechanics and Analysis, 3 (1954), 13.

[30]

P. Huber, Dünngitter-Spektralmethoden zur Approximation des Frobenius-Perron-Operators,, Diploma thesis (in German), (2009).

[31]

M. R. Jovanović, P. J. Schmid and J. W. Nichols, Sparsity-promoting dynamic mode decomposition,, Physics of Fluids, 26 ().

[32]

O. Junge and P. Koltai, Discretization of the Frobenius-Perron operator using a sparse Haar tensor basis: The Sparse Ulam method,, SIAM Journal on Numerical Analysis, 47 (2009), 3464. doi: 10.1137/080716864.

[33]

P. Koltai, Efficient Approximation Methods for the Global Long-Term Behavior of Dynamical Systems - Theory, Algorithms and Examples,, PhD thesis, (2010).

[34]

B. O. Koopman, Hamiltonian systems and transformation in Hilbert space,, Proceedings of the National Academy of Sciences of the United States of America, 17 (1931), 315. doi: 10.1073/pnas.17.5.315.

[35]

U. Krengel, Ergodic Theorems, vol. 6 of de Gruyter Studies in Mathematics,, Walter de Gruyter & Co., (1985). doi: 10.1515/9783110844641.

[36]

A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, vol. 97 of Applied Mathematical Sciences,, 2nd edition, (1994). doi: 10.1007/978-1-4612-4286-4.

[37]

T.-Y. Li, Finite approximation for the Frobenius-Perron operator. A solution to Ulam's conjecture,, Journal of Approximation Theory, 17 (1976), 177. doi: 10.1016/0021-9045(76)90037-X.

[38]

J. C. Mattingly and A. M. Stuart, Geometric ergodicity of some hypo-elliptic diffusions for particle motions,, Markov Process. Related Fields, 8 (2002), 199.

[39]

S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability,, Springer Science & Business Media, (2012).

[40]

R. Murray, Discrete Approximation of Invariant Densities,, PhD thesis, (1997).

[41]

R. Murray, Optimal partition choice for invariant measure approximation for one-dimensional maps,, Nonlinearity, 17 (2004), 1623. doi: 10.1088/0951-7715/17/5/004.

[42]

F. Noé and F. Nüske, A variational approach to modeling slow processes in stochastic dynamical systems,, Multiscale Modeling & Simulation, 11 (2013), 635. doi: 10.1137/110858616.

[43]

F. Nüske, B. G. Keller, G. Pérez-Hernández, A. S. J. S. Mey and F. Noé, Variational approach to molecular kinetics,, Journal of Chemical Theory and Computation, 10 (2014), 1739.

[44]

F. Nüske, R. Schneider, F. Vitalini and F. Noé, Variational tensor approach for approximating the rare-event kinetics of macromolecular systems,, The Journal of Chemical Physics, 144 ().

[45]

S. Ober-Blöbaum and K. Padberg-Gehle, Multiobjective optimal control of fluid mixing,, PAMM, 15 (2015), 639.

[46]

D. Ornstein, Bernoulli shifts with the same entropy are isomorphic,, Advances in Mathematics, 4 (1970), 337. doi: 10.1016/0001-8708(70)90029-0.

[47]

R. Preis, M. Dellnitz, M. Hessel, C. Schütte and E. Meerbach, Dominant Paths Between Almost Invariant Sets of Dynamical Systems,, DFG Schwerpunktprogramm 1095, (1095).

[48]

P. Schmid and J. Sesterhenn, Dynamic mode decomposition of numerical and experimental data,, in 61st Annual Meeting of the APS Division of Fluid Dynamics, (2008).

[49]

Schrödinger, LLC, The PyMOL molecular graphics system,, Version 1.7.4, (2014).

[50]

C. Schütte, Conformational Dynamics: Modelling, Theory, Algorithm, and Application to Biomolecules, 1999,, Habilitation Thesis., ().

[51]

C. Schütte and M. Sarich, Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches,, no. 24 in Courant Lecture Notes, (2013).

[52]

Y. G. Sinai, On the notion of entropy of dynamical systems,, in Doklady Akademii Nauk, 124 (1959), 768.

[53]

A. Tantet, V. Lucarini, F. Lunkeit and H. A. Dijkstra, Crisis of the chaotic attractor of a climate model: A transfer operator approach,, Preprint, ().

[54]

A. Tantet, F. R. van der Burgt and H. A. Dijkstra, An early warning indicator for atmospheric blocking events using transfer operators,, Chaos, 25 (2015). doi: 10.1063/1.4908174.

[55]

J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton and J. N. Kutz, On dynamic mode decomposition: Theory and applications,, ArXiv e-prints., ().

[56]

S. M. Ulam, A Collection of Mathematical Problems,, Interscience Publisher NY, (1960).

[57]

U. Vaidya, P. G. Mehta and U. V. Shanbhag, Nonlinear stabilization via control Lyapunov measure,, IEEE Transactions on Automatic Control, 55 (2010), 1314. doi: 10.1109/TAC.2010.2042226.

[58]

M. O. Williams, I. G. Kevrekidis and C. W. Rowley, A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition,, J. Nonlinear Sci., 25 (2015), 1307. doi: 10.1007/s00332-015-9258-5.

[59]

M. O. Williams, C. W. Rowley and I. G. Kevrekidis, A kernel-based approach to data-driven Koopman spectral analysis,, J. Comput. Dyn., 2 (2015), 247. doi: 10.3934/jcd.2015005.

[60]

M. O. Williams, I. I. Rypina and C. W. Rowley, Identifying finite-time coherent sets from limited quantities of Lagrangian data,, Chaos, 25 (2015). doi: 10.1063/1.4927424.

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