• Previous Article
    Asymptotic invariance and the discretisation of nonautonomous forward attracting sets
  • JCD Home
  • This Issue
  • Next Article
    Towards tensor-based methods for the numerical approximation of the Perron--Frobenius and Koopman operator
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,, , ().   Google Scholar

[2]

Second edition. Dover Publications, Inc., Mineola, NY, 2001.  Google Scholar

[3]

Journal of Computational Physics, 176 (2002), 430-455. doi: 10.1006/jcph.2002.6995.  Google Scholar

[4]

in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (ed. B. Fiedler), Springer, 2001, 145-174, 805-807.  Google Scholar

[5]

SIAM Journal on Numerical Analysis, 36 (1999), 491-515. doi: 10.1137/S0036142996313002.  Google Scholar

[6]

Graduate studies in mathematics, American Mathematical Society, 2010. doi: 10.1090/gsm/019.  Google Scholar

[7]

Physica D, 239 (2010), 1527-1541. doi: 10.1016/j.physd.2010.03.009.  Google Scholar

[8]

Physica D, 238 (2009), 1507-1523. doi: 10.1016/j.physd.2009.03.002.  Google Scholar

[9]

Chaos, 20 (2010), 043116, 10pp. doi: 10.1063/1.3502450.  Google Scholar

[10]

Physica D: Nonlinear Phenomena, 250 (2013), 1-19. doi: 10.1016/j.physd.2013.01.013.  Google Scholar

[11]

SIAM Journal on Scientific Computing, 24 (2003), 1839-1863. doi: 10.1137/S106482750238911X.  Google Scholar

[12]

Chaos: An Interdisciplinary Journal of Nonlinear Science, 25 (2015), 087409, 11pp. doi: 10.1063/1.4927640.  Google Scholar

[13]

SIAM Journal on Numerical Analysis, 51 (2013), 223-247. doi: 10.1137/110819986.  Google Scholar

[14]

G. Froyland and P. Koltai, Estimating long-term behavior of periodically driven flows without trajectory integration,, , ().   Google Scholar

[15]

in Ergodic Theory, Open Dynamics, and Coherent Structures, Springer, 70 (2014), 171-216. doi: 10.1007/978-1-4939-0419-8_9.  Google Scholar

[16]

Phys. Rev. E, 93 (2016), 063107. doi: 10.1103/PhysRevE.93.063107.  Google Scholar

[17]

Physica D, 149 (2001), 248-277. doi: 10.1016/S0167-2789(00)00199-8.  Google Scholar

[18]

Physica D, 240 (2011), 574-598. doi: 10.1016/j.physd.2010.11.010.  Google Scholar

[19]

Physica D, 147 (2000), 352-370. doi: 10.1016/S0167-2789(00)00142-1.  Google Scholar

[20]

in New Algorithms for Macromolecular Simulation, Springer, 49 (2006), 167-182. doi: 10.1007/3-540-31618-3_11.  Google Scholar

[21]

in Proceedings of the 43rd IEEE Conference on Decision and Control, 2 (2004), 2225-2230. doi: 10.1109/CDC.2004.1430379.  Google Scholar

[22]

SIAM Journal on Scientific Computing, 26 (2005), 1214-1233. doi: 10.1137/S1064827502410633.  Google Scholar

[23]

Second edition. Applied Mathematical Sciences, 97. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4286-4.  Google Scholar

[24]

J. Approx. Theory, 17 (1976), 177-186. doi: 10.1016/0021-9045(76)90037-X.  Google Scholar

[25]

2008, (Massachusetts Institute of Technology: MIT OpenCouseWare), http://ocw.mit.edu (Accessed July 13, 2015). License: Creative Commons BY-NC-SA. Google Scholar

[26]

B. Oksendal, Stochastic Differential Equations: An Introduction with Applications,, Springer, ().   Google Scholar

[27]

Journal of Computational Physics, 151 (1999), 146-168. doi: 10.1006/jcph.1999.6231.  Google Scholar

[28]

Courier Dover Publications, 2004. Google Scholar

[29]

Applied Numerical Mathematics, 7 (1991), 27-40. doi: 10.1016/0168-9274(91)90102-6.  Google Scholar

show all references

References:
[1]

R. Banisch and P. Koltai, Understanding the geometry of transport: diffusion maps for Lagrangian trajectory data unravel coherent sets,, , ().   Google Scholar

[2]

Second edition. Dover Publications, Inc., Mineola, NY, 2001.  Google Scholar

[3]

Journal of Computational Physics, 176 (2002), 430-455. doi: 10.1006/jcph.2002.6995.  Google Scholar

[4]

in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (ed. B. Fiedler), Springer, 2001, 145-174, 805-807.  Google Scholar

[5]

SIAM Journal on Numerical Analysis, 36 (1999), 491-515. doi: 10.1137/S0036142996313002.  Google Scholar

[6]

Graduate studies in mathematics, American Mathematical Society, 2010. doi: 10.1090/gsm/019.  Google Scholar

[7]

Physica D, 239 (2010), 1527-1541. doi: 10.1016/j.physd.2010.03.009.  Google Scholar

[8]

Physica D, 238 (2009), 1507-1523. doi: 10.1016/j.physd.2009.03.002.  Google Scholar

[9]

Chaos, 20 (2010), 043116, 10pp. doi: 10.1063/1.3502450.  Google Scholar

[10]

Physica D: Nonlinear Phenomena, 250 (2013), 1-19. doi: 10.1016/j.physd.2013.01.013.  Google Scholar

[11]

SIAM Journal on Scientific Computing, 24 (2003), 1839-1863. doi: 10.1137/S106482750238911X.  Google Scholar

[12]

Chaos: An Interdisciplinary Journal of Nonlinear Science, 25 (2015), 087409, 11pp. doi: 10.1063/1.4927640.  Google Scholar

[13]

SIAM Journal on Numerical Analysis, 51 (2013), 223-247. doi: 10.1137/110819986.  Google Scholar

[14]

G. Froyland and P. Koltai, Estimating long-term behavior of periodically driven flows without trajectory integration,, , ().   Google Scholar

[15]

in Ergodic Theory, Open Dynamics, and Coherent Structures, Springer, 70 (2014), 171-216. doi: 10.1007/978-1-4939-0419-8_9.  Google Scholar

[16]

Phys. Rev. E, 93 (2016), 063107. doi: 10.1103/PhysRevE.93.063107.  Google Scholar

[17]

Physica D, 149 (2001), 248-277. doi: 10.1016/S0167-2789(00)00199-8.  Google Scholar

[18]

Physica D, 240 (2011), 574-598. doi: 10.1016/j.physd.2010.11.010.  Google Scholar

[19]

Physica D, 147 (2000), 352-370. doi: 10.1016/S0167-2789(00)00142-1.  Google Scholar

[20]

in New Algorithms for Macromolecular Simulation, Springer, 49 (2006), 167-182. doi: 10.1007/3-540-31618-3_11.  Google Scholar

[21]

in Proceedings of the 43rd IEEE Conference on Decision and Control, 2 (2004), 2225-2230. doi: 10.1109/CDC.2004.1430379.  Google Scholar

[22]

SIAM Journal on Scientific Computing, 26 (2005), 1214-1233. doi: 10.1137/S1064827502410633.  Google Scholar

[23]

Second edition. Applied Mathematical Sciences, 97. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4286-4.  Google Scholar

[24]

J. Approx. Theory, 17 (1976), 177-186. doi: 10.1016/0021-9045(76)90037-X.  Google Scholar

[25]

2008, (Massachusetts Institute of Technology: MIT OpenCouseWare), http://ocw.mit.edu (Accessed July 13, 2015). License: Creative Commons BY-NC-SA. Google Scholar

[26]

B. Oksendal, Stochastic Differential Equations: An Introduction with Applications,, Springer, ().   Google Scholar

[27]

Journal of Computational Physics, 151 (1999), 146-168. doi: 10.1006/jcph.1999.6231.  Google Scholar

[28]

Courier Dover Publications, 2004. Google Scholar

[29]

Applied Numerical Mathematics, 7 (1991), 27-40. doi: 10.1016/0168-9274(91)90102-6.  Google Scholar

[1]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250

[2]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017

[3]

Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic & Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016

[4]

Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485

[5]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401

[6]

Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056

[7]

Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028

[8]

Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028

[9]

Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215

[10]

Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic & Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009

[11]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165

[12]

Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021013

[13]

Ludovic Dan Lemle. $L^1(R^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with a class of Dirichlet operators. Electronic Research Announcements, 2008, 15: 65-70. doi: 10.3934/era.2008.15.65

[14]

Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079

[15]

Simon Plazotta. A BDF2-approach for the non-linear Fokker-Planck equation. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2893-2913. doi: 10.3934/dcds.2019120

[16]

Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic & Related Models, 2019, 12 (4) : 727-748. doi: 10.3934/krm.2019028

[17]

Florian Schneider, Andreas Roth, Jochen Kall. First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions. Kinetic & Related Models, 2017, 10 (4) : 1127-1161. doi: 10.3934/krm.2017044

[18]

Zeinab Karaki. Trend to the equilibrium for the Fokker-Planck system with an external magnetic field. Kinetic & Related Models, 2020, 13 (2) : 309-344. doi: 10.3934/krm.2020011

[19]

Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 845-870. doi: 10.3934/cpaa.2009.8.845

[20]

Yuan Gao, Guangzhen Jin, Jian-Guo Liu. Inbetweening auto-animation via Fokker-Planck dynamics and thresholding. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021016

 Impact Factor: 

Metrics

  • PDF downloads (150)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]