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June  2016, 3(2): 163-177. doi: 10.3934/jcd.2016008

## Computing coherent sets using the Fokker-Planck equation

 1 Center for Mathematics, Technische Universität München, 85747 Garching bei München, Germany, Germany

Received  December 2015 Revised  October 2016 Published  December 2016

We perform a numerical approximation of coherent sets in finite-dimensional smooth dynamical systems by computing singular vectors of the transfer operator for a stochastically perturbed flow. This operator is obtained by solution of a discretized Fokker-Planck equation. For numerical implementation, we employ spectral collocation methods and an exponential time differentiation scheme. We experimentally compare our approach with the more classical method by Ulam that is based on discretization of the transfer operator of the unperturbed flow.
Citation: Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008
##### References:
 [1] R. Banisch and P. Koltai, Understanding the geometry of transport: diffusion maps for Lagrangian trajectory data unravel coherent sets,, , (). [2] J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second edition. Dover Publications, Inc., Mineola, NY, 2001. [3] S. Cox and P. Matthews, Exponential time differencing for stiff systems, Journal of Computational Physics, 176 (2002), 430-455. doi: 10.1006/jcph.2002.6995. [4] 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 (ed. B. Fiedler), Springer, 2001, 145-174, 805-807. [5] 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. [6] L. Evans, Partial Differential Equations, Graduate studies in mathematics, American Mathematical Society, 2010. doi: 10.1090/gsm/019. [7] G. Froyland, S. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems, Physica D, 239 (2010), 1527-1541. doi: 10.1016/j.physd.2010.03.009. [8] G. Froyland and K. Padberg, Almost-invariant sets and invariant manifolds - connecting probabilistic and geometric descriptions of coherent structures in flows, Physica D, 238 (2009), 1507-1523. doi: 10.1016/j.physd.2009.03.002. [9] G. Froyland, N. Santitissadeekorn and A. Monahan, Transport in time-dependent dynamical systems: Finite-time coherent sets, Chaos, 20 (2010), 043116, 10pp. doi: 10.1063/1.3502450. [10] G. Froyland, An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems, Physica D: Nonlinear Phenomena, 250 (2013), 1-19. doi: 10.1016/j.physd.2013.01.013. [11] G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost-invariant sets and cycles, SIAM Journal on Scientific Computing, 24 (2003), 1839-1863. doi: 10.1137/S106482750238911X. [12] G. Froyland and O. Junge, On fast computation of finite-time coherent sets using radial basis functions, Chaos: An Interdisciplinary Journal of Nonlinear Science, 25 (2015), 087409, 11pp. doi: 10.1063/1.4927640. [13] 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), 223-247. doi: 10.1137/110819986. [14] G. Froyland and P. Koltai, Estimating long-term behavior of periodically driven flows without trajectory integration,, , (). [15] G. Froyland and K. Padberg-Gehle, Almost-invariant and finite-time coherent sets: Directionality, duration, and diffusion, in Ergodic Theory, Open Dynamics, and Coherent Structures, Springer, 70 (2014), 171-216. doi: 10.1007/978-1-4939-0419-8_9. [16] A. Hadjighasem, D. Karrasch, H. Teramoto and G. Haller, Spectral-clustering approach to lagrangian vortex detection, Phys. Rev. E, 93 (2016), 063107. doi: 10.1103/PhysRevE.93.063107. [17] G. Haller, Distinguished material surfaces and coherent structures in three-dimensional fluid flows, Physica D, 149 (2001), 248-277. doi: 10.1016/S0167-2789(00)00199-8. [18] G. Haller, A variational theory of hyperbolic Lagrangian coherent structures, Physica D, 240 (2011), 574-598. doi: 10.1016/j.physd.2010.11.010. [19] G. Haller and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulence, Physica D, 147 (2000), 352-370. doi: 10.1016/S0167-2789(00)00142-1. [20] W. Huisinga and B. Schmidt, Metastability and dominant eigenvalues of transfer operators, in New Algorithms for Macromolecular Simulation, Springer, 49 (2006), 167-182. doi: 10.1007/3-540-31618-3_11. [21] O. Junge, J. E. Marsden and I. Mezic, Uncertainty in the dynamics of conservative maps, in Proceedings of the 43rd IEEE Conference on Decision and Control, 2 (2004), 2225-2230. doi: 10.1109/CDC.2004.1430379. [22] A.-K. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff pdes, SIAM Journal on Scientific Computing, 26 (2005), 1214-1233. doi: 10.1137/S1064827502410633. [23] A. Lasota and M. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, Second edition. Applied Mathematical Sciences, 97. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4286-4. [24] T. Y. Li, Finite Approximation for the Frobenius-Perron Operator. A Solution to Ulam's Conjecture, J. Approx. Theory, 17 (1976), 177-186. doi: 10.1016/0021-9045(76)90037-X. [25] J.-C. Nave, Computational Science and Engineering, 2008, (Massachusetts Institute of Technology: MIT OpenCouseWare), http://ocw.mit.edu (Accessed July 13, 2015). License: Creative Commons BY-NC-SA. [26] B. Oksendal, Stochastic Differential Equations: An Introduction with Applications,, Springer, (). [27] C. Schütte, A. Fischer, W. Huisinga and P. Deuflhard, A direct approach to conformational dynamics based on hybrid monte carlo, Journal of Computational Physics, 151 (1999), 146-168. doi: 10.1006/jcph.1999.6231. [28] S. M. Ulam, Problems in Modern Mathematics, Courier Dover Publications, 2004. [29] T. A. Zang, On the rotation and skew-symmetric forms for incompressible flow simulations, Applied Numerical Mathematics, 7 (1991), 27-40. doi: 10.1016/0168-9274(91)90102-6.

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##### References:
 [1] R. Banisch and P. Koltai, Understanding the geometry of transport: diffusion maps for Lagrangian trajectory data unravel coherent sets,, , (). [2] J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second edition. Dover Publications, Inc., Mineola, NY, 2001. [3] S. Cox and P. Matthews, Exponential time differencing for stiff systems, Journal of Computational Physics, 176 (2002), 430-455. doi: 10.1006/jcph.2002.6995. [4] 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 (ed. B. Fiedler), Springer, 2001, 145-174, 805-807. [5] 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. [6] L. Evans, Partial Differential Equations, Graduate studies in mathematics, American Mathematical Society, 2010. doi: 10.1090/gsm/019. [7] G. Froyland, S. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems, Physica D, 239 (2010), 1527-1541. doi: 10.1016/j.physd.2010.03.009. [8] G. Froyland and K. Padberg, Almost-invariant sets and invariant manifolds - connecting probabilistic and geometric descriptions of coherent structures in flows, Physica D, 238 (2009), 1507-1523. doi: 10.1016/j.physd.2009.03.002. [9] G. Froyland, N. Santitissadeekorn and A. Monahan, Transport in time-dependent dynamical systems: Finite-time coherent sets, Chaos, 20 (2010), 043116, 10pp. doi: 10.1063/1.3502450. [10] G. Froyland, An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems, Physica D: Nonlinear Phenomena, 250 (2013), 1-19. doi: 10.1016/j.physd.2013.01.013. [11] G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost-invariant sets and cycles, SIAM Journal on Scientific Computing, 24 (2003), 1839-1863. doi: 10.1137/S106482750238911X. [12] G. Froyland and O. Junge, On fast computation of finite-time coherent sets using radial basis functions, Chaos: An Interdisciplinary Journal of Nonlinear Science, 25 (2015), 087409, 11pp. doi: 10.1063/1.4927640. [13] 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), 223-247. doi: 10.1137/110819986. [14] G. Froyland and P. Koltai, Estimating long-term behavior of periodically driven flows without trajectory integration,, , (). [15] G. Froyland and K. Padberg-Gehle, Almost-invariant and finite-time coherent sets: Directionality, duration, and diffusion, in Ergodic Theory, Open Dynamics, and Coherent Structures, Springer, 70 (2014), 171-216. doi: 10.1007/978-1-4939-0419-8_9. [16] A. Hadjighasem, D. Karrasch, H. Teramoto and G. Haller, Spectral-clustering approach to lagrangian vortex detection, Phys. Rev. E, 93 (2016), 063107. doi: 10.1103/PhysRevE.93.063107. [17] G. Haller, Distinguished material surfaces and coherent structures in three-dimensional fluid flows, Physica D, 149 (2001), 248-277. doi: 10.1016/S0167-2789(00)00199-8. [18] G. Haller, A variational theory of hyperbolic Lagrangian coherent structures, Physica D, 240 (2011), 574-598. doi: 10.1016/j.physd.2010.11.010. [19] G. Haller and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulence, Physica D, 147 (2000), 352-370. doi: 10.1016/S0167-2789(00)00142-1. [20] W. Huisinga and B. Schmidt, Metastability and dominant eigenvalues of transfer operators, in New Algorithms for Macromolecular Simulation, Springer, 49 (2006), 167-182. doi: 10.1007/3-540-31618-3_11. [21] O. Junge, J. E. Marsden and I. Mezic, Uncertainty in the dynamics of conservative maps, in Proceedings of the 43rd IEEE Conference on Decision and Control, 2 (2004), 2225-2230. doi: 10.1109/CDC.2004.1430379. [22] A.-K. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff pdes, SIAM Journal on Scientific Computing, 26 (2005), 1214-1233. doi: 10.1137/S1064827502410633. [23] A. Lasota and M. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, Second edition. Applied Mathematical Sciences, 97. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4286-4. [24] T. Y. Li, Finite Approximation for the Frobenius-Perron Operator. A Solution to Ulam's Conjecture, J. Approx. Theory, 17 (1976), 177-186. doi: 10.1016/0021-9045(76)90037-X. [25] J.-C. Nave, Computational Science and Engineering, 2008, (Massachusetts Institute of Technology: MIT OpenCouseWare), http://ocw.mit.edu (Accessed July 13, 2015). License: Creative Commons BY-NC-SA. [26] B. Oksendal, Stochastic Differential Equations: An Introduction with Applications,, Springer, (). [27] C. Schütte, A. Fischer, W. Huisinga and P. Deuflhard, A direct approach to conformational dynamics based on hybrid monte carlo, Journal of Computational Physics, 151 (1999), 146-168. doi: 10.1006/jcph.1999.6231. [28] S. M. Ulam, Problems in Modern Mathematics, Courier Dover Publications, 2004. [29] T. A. Zang, On the rotation and skew-symmetric forms for incompressible flow simulations, Applied Numerical Mathematics, 7 (1991), 27-40. doi: 10.1016/0168-9274(91)90102-6.
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