American Institute of Mathematical Sciences

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June  2016, 3(2): 139-161. doi: 10.3934/jcd.2016007

Towards tensor-based methods for the numerical approximation of the Perron--Frobenius and Koopman operator

Stefan Klus 1, and  Christof Schütte 1,

1. 

Department of Mathematics and Computer Science, Freie Universität Berlin

Received  November 2015 Revised  August 2016 Published  November 2016

The global behavior of dynamical systems can be studied by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with the system. Two important operators which are frequently used to gain insight into the system's behavior are the Perron--Frobenius operator and the Koopman operator. Due to the curse of dimensionality, computing the eigenfunctions of high-dimensional systems is in general infeasible. We will propose a tensor-based reformulation of two numerical methods for computing finite-dimensional approximations of the aforementioned infinite-dimensional operators, namely Ulam's method and Extended Dynamic Mode Decomposition (EDMD). The aim of the tensor formulation is to approximate the eigenfunctions by low-rank tensors, potentially resulting in a significant reduction of the time and memory required to solve the resulting eigenvalue problems, provided that such a low-rank tensor decomposition exists. Typically, not all variables of a high-dimensional dynamical system contribute equally to the system's behavior, often the dynamics can be decomposed into slow and fast processes, which is also reflected in the eigenfunctions. Thus, the weak coupling between different variables might be approximated by low-rank tensor cores. We will illustrate the efficiency of the tensor-based formulation of Ulam's method and EDMD using simple stochastic differential equations.
Keywords: Perron--Frobenius operator, TT format., Ulam's method, Koopman operator, extended dynamic mode decomposition.
Mathematics Subject Classification: Primary: 37N25, 37L65, 15A6.
Citation: Stefan Klus, Christof Schütte. Towards tensor-based methods for the numerical approximation of the Perron--Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (2) : 139-161. doi: 10.3934/jcd.2016007
References:
[1]

G. Beylkin and M. J. Mohlenkamp, Numerical operator calculus in higher dimensions, Proceedings of the National Academy of Sciences, 99 (2002), 10246-10251. doi: 10.1073/pnas.112329799.  Google Scholar

[2]

G. Beylkin and M. J. Mohlenkamp, Algorithms for numerical analysis in high dimensions, SIAM Journal on Scientific Computing, 26 (2005), 2133-2159. doi: 10.1137/040604959.  Google Scholar

[3]

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

[4]

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

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G. Chen and T. Ueta (eds.), Chaos in Circuits and Systems, World Scientific Series on Nonlinear Science, Series B, Volume 11, World Scientific, 2002. doi: 10.1142/9789812705303.  Google Scholar

[6]

M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - Set oriented numerical methods for dynamical systems, in Ergodic theory, analysis, and efficient simulation of dynamical systems, Springer, 2001, 145-174.  Google Scholar

[7]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM Journal on Numerical Analysis, 36 (1999), 491-515. doi: 10.1137/S0036142996313002.  Google Scholar

[8]

J. Ding, Q. Du and T.-Y. Li, High order approximation of the Frobenius-Perron operator, Applied Mathematics and Computation, 53 (1993), 151-171. doi: 10.1016/0096-3003(93)90099-Z.  Google Scholar

[9]

G. Friesecke, O. Junge and P. Koltai, Mean field approximation in conformation dynamics, Multiscale Modeling & Simulation, 8 (2009), 254-268. doi: 10.1137/080745262.  Google Scholar

[10]

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), 1816-1846. doi: 10.1137/130943637.  Google Scholar

[11]

G. Froyland, G. A. Gottwald and A. Hammerlindl, A trajectory-free framework for analysing multiscale systems,, ArXiv e-prints., ().   Google Scholar

[12]

P. Gelß, S. Matera and C. Schütte, Solving the master equation without kinetic Monte Carlo: Tensor train approximations for a CO oxidation model, Journal of Computational Physics, 314 (2016), 489-502. doi: 10.1016/j.jcp.2016.03.025.  Google Scholar

[13]

G. H. Golub and C. F. V. Loan, Matrix Computations, 4th edition, Johns Hopkins University Press, 2013.  Google Scholar

[14]

L. Grasedyck, D. Kressner and C. Tobler, A literature survey of low-rank tensor approximation techniques, GAMM-Mitt., 36 (2013), 53-78. doi: 10.1002/gamm.201310004.  Google Scholar

[15]

W. Hackbusch, Numerical tensor calculus, Acta Numerica, 23 (2014), 651-742. doi: 10.1017/S0962492914000087.  Google Scholar

[16]

S. Holtz, T. Rohwedder and R. Schneider, The alternating linear scheme for tensor optimization in the tensor train format, SIAM Journal on Scientific Computing, 34 (2012), A683-A713. doi: 10.1137/100818893.  Google Scholar

[17]

S. Klus, P. Koltai and C. Schütte, On the numerical approximation of the Perron-Frobenius and Koopman operator, Journal of Computational Dynamics, 3 (2016), 51-79. doi: 10.3934/jcd.2016003.  Google Scholar

[18]

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), 1739-1752. Google Scholar

[19]

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 ().   Google Scholar

[20]

I. V. Oseledets, Tensor-train decomposition, SIAM Journal on Scientific Computing, 33 (2011), 2295-2317. doi: 10.1137/090752286.  Google Scholar

[21]

I. V. Oseledets, TT-toolbox 2.0: Fast multidimensional Array Operations in TT Format,, 2011., ().   Google Scholar

[22]

R. Preis, M. Dellnitz, M. Hessel, C. Schütte and E. Meerbach, Dominant Paths Between Almost Invariant Sets of Dynamical Systems, DFG Schwerpunktprogramm 1095, Technical Report 154, 2004. Google Scholar

[23]

C. Schütte and M. Sarich, Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches, no. 24 in Courant Lecture Notes, American Mathematical Society, 2013. doi: 10.1090/cln/024.  Google Scholar

[24]

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), 1307-1346. doi: 10.1007/s00332-015-9258-5.  Google Scholar

[25]

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), 247-265. doi: 10.3934/jcd.2015005.  Google Scholar

show all references

References:
[1]

G. Beylkin and M. J. Mohlenkamp, Numerical operator calculus in higher dimensions, Proceedings of the National Academy of Sciences, 99 (2002), 10246-10251. doi: 10.1073/pnas.112329799.  Google Scholar

[2]

G. Beylkin and M. J. Mohlenkamp, Algorithms for numerical analysis in high dimensions, SIAM Journal on Scientific Computing, 26 (2005), 2133-2159. doi: 10.1137/040604959.  Google Scholar

[3]

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

[4]

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

[5]

G. Chen and T. Ueta (eds.), Chaos in Circuits and Systems, World Scientific Series on Nonlinear Science, Series B, Volume 11, World Scientific, 2002. doi: 10.1142/9789812705303.  Google Scholar

[6]

M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - Set oriented numerical methods for dynamical systems, in Ergodic theory, analysis, and efficient simulation of dynamical systems, Springer, 2001, 145-174.  Google Scholar

[7]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM Journal on Numerical Analysis, 36 (1999), 491-515. doi: 10.1137/S0036142996313002.  Google Scholar

[8]

J. Ding, Q. Du and T.-Y. Li, High order approximation of the Frobenius-Perron operator, Applied Mathematics and Computation, 53 (1993), 151-171. doi: 10.1016/0096-3003(93)90099-Z.  Google Scholar

[9]

G. Friesecke, O. Junge and P. Koltai, Mean field approximation in conformation dynamics, Multiscale Modeling & Simulation, 8 (2009), 254-268. doi: 10.1137/080745262.  Google Scholar

[10]

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), 1816-1846. doi: 10.1137/130943637.  Google Scholar

[11]

G. Froyland, G. A. Gottwald and A. Hammerlindl, A trajectory-free framework for analysing multiscale systems,, ArXiv e-prints., ().   Google Scholar

[12]

P. Gelß, S. Matera and C. Schütte, Solving the master equation without kinetic Monte Carlo: Tensor train approximations for a CO oxidation model, Journal of Computational Physics, 314 (2016), 489-502. doi: 10.1016/j.jcp.2016.03.025.  Google Scholar

[13]

G. H. Golub and C. F. V. Loan, Matrix Computations, 4th edition, Johns Hopkins University Press, 2013.  Google Scholar

[14]

L. Grasedyck, D. Kressner and C. Tobler, A literature survey of low-rank tensor approximation techniques, GAMM-Mitt., 36 (2013), 53-78. doi: 10.1002/gamm.201310004.  Google Scholar

[15]

W. Hackbusch, Numerical tensor calculus, Acta Numerica, 23 (2014), 651-742. doi: 10.1017/S0962492914000087.  Google Scholar

[16]

S. Holtz, T. Rohwedder and R. Schneider, The alternating linear scheme for tensor optimization in the tensor train format, SIAM Journal on Scientific Computing, 34 (2012), A683-A713. doi: 10.1137/100818893.  Google Scholar

[17]

S. Klus, P. Koltai and C. Schütte, On the numerical approximation of the Perron-Frobenius and Koopman operator, Journal of Computational Dynamics, 3 (2016), 51-79. doi: 10.3934/jcd.2016003.  Google Scholar

[18]

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), 1739-1752. Google Scholar

[19]

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 ().   Google Scholar

[20]

I. V. Oseledets, Tensor-train decomposition, SIAM Journal on Scientific Computing, 33 (2011), 2295-2317. doi: 10.1137/090752286.  Google Scholar

[21]

I. V. Oseledets, TT-toolbox 2.0: Fast multidimensional Array Operations in TT Format,, 2011., ().   Google Scholar

[22]

R. Preis, M. Dellnitz, M. Hessel, C. Schütte and E. Meerbach, Dominant Paths Between Almost Invariant Sets of Dynamical Systems, DFG Schwerpunktprogramm 1095, Technical Report 154, 2004. Google Scholar

[23]

C. Schütte and M. Sarich, Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches, no. 24 in Courant Lecture Notes, American Mathematical Society, 2013. doi: 10.1090/cln/024.  Google Scholar

[24]

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), 1307-1346. doi: 10.1007/s00332-015-9258-5.  Google Scholar

[25]

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), 247-265. doi: 10.3934/jcd.2015005.  Google Scholar

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