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Towards tensor-based methods for the numerical approximation of the Perron--Frobenius and Koopman operator
Computing coherent sets using the Fokker-Planck equation
1. | Center for Mathematics, Technische Universität München, 85747 Garching bei München, Germany, Germany |
References:
[1] |
R. Banisch and P. Koltai, Understanding the geometry of transport: diffusion maps for Lagrangian trajectory data unravel coherent sets,, , (). Google Scholar |
[2] |
J. P. Boyd, Chebyshev and Fourier Spectral Methods,, Second edition. Dover Publications, (2001).
|
[3] |
S. Cox and P. Matthews, Exponential time differencing for stiff systems,, Journal of Computational Physics, 176 (2002), 430.
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, (2001), 145.
|
[5] |
M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM Journal on Numerical Analysis, 36 (1999), 491.
doi: 10.1137/S0036142996313002. |
[6] |
L. Evans, Partial Differential Equations,, Graduate studies in mathematics, (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.
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.
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).
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.
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.
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).
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.
doi: 10.1137/110819986. |
[14] |
G. Froyland and P. Koltai, Estimating long-term behavior of periodically driven flows without trajectory integration,, , (). Google Scholar |
[15] |
G. Froyland and K. Padberg-Gehle, Almost-invariant and finite-time coherent sets: Directionality, duration, and diffusion,, in Ergodic Theory, 70 (2014), 171.
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).
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.
doi: 10.1016/S0167-2789(00)00199-8. |
[18] |
G. Haller, A variational theory of hyperbolic Lagrangian coherent structures,, Physica D, 240 (2011), 574.
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.
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, 49 (2006), 167.
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.
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.
doi: 10.1137/S1064827502410633. |
[23] |
A. Lasota and M. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics,, Second edition. Applied Mathematical Sciences, (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.
doi: 10.1016/0021-9045(76)90037-X. |
[25] |
J.-C. Nave, Computational Science and Engineering,, 2008, (2015). Google Scholar |
[26] |
B. Oksendal, Stochastic Differential Equations: An Introduction with Applications,, Springer, (). Google Scholar |
[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.
doi: 10.1006/jcph.1999.6231. |
[28] |
S. M. Ulam, Problems in Modern Mathematics,, Courier Dover Publications, (2004). Google Scholar |
[29] |
T. A. Zang, On the rotation and skew-symmetric forms for incompressible flow simulations,, Applied Numerical Mathematics, 7 (1991), 27.
doi: 10.1016/0168-9274(91)90102-6. |
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] |
J. P. Boyd, Chebyshev and Fourier Spectral Methods,, Second edition. Dover Publications, (2001).
|
[3] |
S. Cox and P. Matthews, Exponential time differencing for stiff systems,, Journal of Computational Physics, 176 (2002), 430.
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, (2001), 145.
|
[5] |
M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM Journal on Numerical Analysis, 36 (1999), 491.
doi: 10.1137/S0036142996313002. |
[6] |
L. Evans, Partial Differential Equations,, Graduate studies in mathematics, (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.
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.
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).
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.
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.
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).
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.
doi: 10.1137/110819986. |
[14] |
G. Froyland and P. Koltai, Estimating long-term behavior of periodically driven flows without trajectory integration,, , (). Google Scholar |
[15] |
G. Froyland and K. Padberg-Gehle, Almost-invariant and finite-time coherent sets: Directionality, duration, and diffusion,, in Ergodic Theory, 70 (2014), 171.
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).
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.
doi: 10.1016/S0167-2789(00)00199-8. |
[18] |
G. Haller, A variational theory of hyperbolic Lagrangian coherent structures,, Physica D, 240 (2011), 574.
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.
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, 49 (2006), 167.
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.
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.
doi: 10.1137/S1064827502410633. |
[23] |
A. Lasota and M. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics,, Second edition. Applied Mathematical Sciences, (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.
doi: 10.1016/0021-9045(76)90037-X. |
[25] |
J.-C. Nave, Computational Science and Engineering,, 2008, (2015). Google Scholar |
[26] |
B. Oksendal, Stochastic Differential Equations: An Introduction with Applications,, Springer, (). Google Scholar |
[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.
doi: 10.1006/jcph.1999.6231. |
[28] |
S. M. Ulam, Problems in Modern Mathematics,, Courier Dover Publications, (2004). Google Scholar |
[29] |
T. A. Zang, On the rotation and skew-symmetric forms for incompressible flow simulations,, Applied Numerical Mathematics, 7 (1991), 27.
doi: 10.1016/0168-9274(91)90102-6. |
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