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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 |
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. |
[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. |
[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, Series B, Volume 11, World Scientific, 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, analysis, and efficient simulation of dynamical systems, Springer, 2001, 145-174. |
[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. |
[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. |
[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. |
[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. |
[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), 489-502.
doi: 10.1016/j.jcp.2016.03.025. |
[13] |
G. H. Golub and C. F. V. Loan, Matrix Computations, 4th edition, Johns Hopkins University Press, 2013. |
[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. |
[15] |
W. Hackbusch, Numerical tensor calculus, Acta Numerica, 23 (2014), 651-742.
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), A683-A713.
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), 51-79.
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), 1739-1752. |
[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), 2295-2317.
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, Technical Report 154, 2004. |
[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. |
[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. |
[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. |
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. |
[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. |
[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, Series B, Volume 11, World Scientific, 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, analysis, and efficient simulation of dynamical systems, Springer, 2001, 145-174. |
[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. |
[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. |
[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. |
[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. |
[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), 489-502.
doi: 10.1016/j.jcp.2016.03.025. |
[13] |
G. H. Golub and C. F. V. Loan, Matrix Computations, 4th edition, Johns Hopkins University Press, 2013. |
[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. |
[15] |
W. Hackbusch, Numerical tensor calculus, Acta Numerica, 23 (2014), 651-742.
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), A683-A713.
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), 51-79.
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), 1739-1752. |
[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), 2295-2317.
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, Technical Report 154, 2004. |
[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. |
[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. |
[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. |
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