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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,, , ().
|
[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. |
show all references
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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