# American Institute of Mathematical Sciences

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

 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.
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. doi: 10.1073/pnas.112329799. [2] G. Beylkin and M. J. Mohlenkamp, Algorithms for numerical analysis in high dimensions,, SIAM Journal on Scientific Computing, 26 (2005), 2133. doi: 10.1137/040604959. [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] M. Budišić, R. Mohr and I. Mezić, Applied koopmanism,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (). [5] G. Chen and T. Ueta (eds.), Chaos in Circuits and Systems,, World Scientific Series on Nonlinear Science, (2002). doi: 10.1142/9789812705303. [6] M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - Set oriented numerical methods for dynamical systems,, in Ergodic theory, (2001), 145. [7] M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM Journal on Numerical Analysis, 36 (1999), 491. doi: 10.1137/S0036142996313002. [8] 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. [9] G. Friesecke, O. Junge and P. Koltai, Mean field approximation in conformation dynamics,, Multiscale Modeling & Simulation, 8 (2009), 254. doi: 10.1137/080745262. [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. doi: 10.1137/130943637. [11] G. Froyland, G. A. Gottwald and A. Hammerlindl, A trajectory-free framework for analysing multiscale systems,, ArXiv e-prints., (). [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. doi: 10.1016/j.jcp.2016.03.025. [13] G. H. Golub and C. F. V. Loan, Matrix Computations,, 4th edition, (2013). [14] L. Grasedyck, D. Kressner and C. Tobler, A literature survey of low-rank tensor approximation techniques,, GAMM-Mitt., 36 (2013), 53. doi: 10.1002/gamm.201310004. [15] W. Hackbusch, Numerical tensor calculus,, Acta Numerica, 23 (2014), 651. doi: 10.1017/S0962492914000087. [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). doi: 10.1137/100818893. [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. doi: 10.3934/jcd.2016003. [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. [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 (). [20] I. V. Oseledets, Tensor-train decomposition,, SIAM Journal on Scientific Computing, 33 (2011), 2295. doi: 10.1137/090752286. [21] I. V. Oseledets, TT-toolbox 2.0: Fast multidimensional Array Operations in TT Format,, 2011., (). [22] R. Preis, M. Dellnitz, M. Hessel, C. Schütte and E. Meerbach, Dominant Paths Between Almost Invariant Sets of Dynamical Systems,, DFG Schwerpunktprogramm 1095, (1095). [23] C. Schütte and M. Sarich, Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches,, no. 24 in Courant Lecture Notes, (2013). doi: 10.1090/cln/024. [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. doi: 10.1007/s00332-015-9258-5. [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. doi: 10.3934/jcd.2015005.

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##### References:
 [1] G. Beylkin and M. J. Mohlenkamp, Numerical operator calculus in higher dimensions,, Proceedings of the National Academy of Sciences, 99 (2002), 10246. doi: 10.1073/pnas.112329799. [2] G. Beylkin and M. J. Mohlenkamp, Algorithms for numerical analysis in high dimensions,, SIAM Journal on Scientific Computing, 26 (2005), 2133. doi: 10.1137/040604959. [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] M. Budišić, R. Mohr and I. Mezić, Applied koopmanism,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (). [5] G. Chen and T. Ueta (eds.), Chaos in Circuits and Systems,, World Scientific Series on Nonlinear Science, (2002). doi: 10.1142/9789812705303. [6] M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - Set oriented numerical methods for dynamical systems,, in Ergodic theory, (2001), 145. [7] M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM Journal on Numerical Analysis, 36 (1999), 491. doi: 10.1137/S0036142996313002. [8] 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. [9] G. Friesecke, O. Junge and P. Koltai, Mean field approximation in conformation dynamics,, Multiscale Modeling & Simulation, 8 (2009), 254. doi: 10.1137/080745262. [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. doi: 10.1137/130943637. [11] G. Froyland, G. A. Gottwald and A. Hammerlindl, A trajectory-free framework for analysing multiscale systems,, ArXiv e-prints., (). [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. doi: 10.1016/j.jcp.2016.03.025. [13] G. H. Golub and C. F. V. Loan, Matrix Computations,, 4th edition, (2013). [14] L. Grasedyck, D. Kressner and C. Tobler, A literature survey of low-rank tensor approximation techniques,, GAMM-Mitt., 36 (2013), 53. doi: 10.1002/gamm.201310004. [15] W. Hackbusch, Numerical tensor calculus,, Acta Numerica, 23 (2014), 651. doi: 10.1017/S0962492914000087. [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). doi: 10.1137/100818893. [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. doi: 10.3934/jcd.2016003. [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. [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 (). [20] I. V. Oseledets, Tensor-train decomposition,, SIAM Journal on Scientific Computing, 33 (2011), 2295. doi: 10.1137/090752286. [21] I. V. Oseledets, TT-toolbox 2.0: Fast multidimensional Array Operations in TT Format,, 2011., (). [22] R. Preis, M. Dellnitz, M. Hessel, C. Schütte and E. Meerbach, Dominant Paths Between Almost Invariant Sets of Dynamical Systems,, DFG Schwerpunktprogramm 1095, (1095). [23] C. Schütte and M. Sarich, Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches,, no. 24 in Courant Lecture Notes, (2013). doi: 10.1090/cln/024. [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. doi: 10.1007/s00332-015-9258-5. [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. doi: 10.3934/jcd.2015005.
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