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On the geometry of twisted prolongations, and dynamical systems
Predicting uncertainty in geometric fluid mechanics
1. | CNRS - LMD - IPSL, École Normale Supérieure de Paris, 24 Rue Lhomond, 75005, Paris, France |
2. | Department of Mathematics, Imperial College, London SW7 2AZ, UK, Springfield, MO 65810, USA |
We review opportunities for stochastic geometric mechanics to incorporate observed data into variational principles, in order to derive data-driven nonlinear dynamical models of effects on the variability of computationally resolvable scales of fluid motion, due to unresolvable, small, rapid scales of fluid motion.
References:
[1] |
S. Albeverio, A. B. Cruzeiro and D. D. Holm, Stochastic Geometric Mechanics, Springer, 2017. |
[2] |
A. Arnaudon, A. L. de Castro and D. D. Holm,
Noise and dissipation on coadjoint orbits, J. Nonlin. Sci., 28 (2018), 91-145.
doi: 10.1007/s00332-017-9404-3. |
[3] |
V. I. Arnol'd,
Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Annales de l'institut Fourier, 16 (1966), 319-361.
doi: 10.5802/aif.233. |
[4] |
V. I. Arnol'd, Mathematical Methods of Classical Mechanics, volume 60 of Graduate Texts in Mathematics, Springer-Verlag, New York |
[5] |
J.-M. Bismut, Mécanique aléatoire, In Tenth Saint Flour Probability Summer School—1980 (Saint Flour, 1980), volume 929 of Lecture Notes in Math., pages 1–100, Springer, Berlin-New York, 1982. |
[6] |
N. Bou-Rabee and H. Owhadi,
Stochastic variational integrators, IMA J. Numer. Anal., 29 (2009), 421-443.
doi: 10.1093/imanum/drn018. |
[7] |
C. J. Cotter, G. A. Gottwald and D. D. Holm, Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics, Proc. Roy. Soc. A, 473 (2017), 20170388, 10pp.
doi: 10.1098/rspa.2017.0388. |
[8] |
C. J. Cotter, D. Crisan, D. D. Holm, W. Pan and I. Shevchenko, Modelling uncertainty using circulation-preserving stochastic transport noise in a 2-layer quasi-geostrophic model, arXiv preprint, arXiv: 1802.05711. |
[9] |
D. Crisan, F. Flandoli and D. Holm, Solution properties of a 3d stochastic Euler fluid equation, arXiv: 1704.06989, [math-ph], 2017.
doi: 10.1007/s00332-018-9506-6. |
[10] |
A. B. Cruzeiro, D. D. Holm and T. S. Ratiu,
Momentum maps and stochastic Clebsch action principles, Commun. in Math. Phys., 357 (2018), 873-912.
doi: 10.1007/s00220-017-3048-x. |
[11] |
F. Gay-Balmaz and D. D. Holm,
Stochastic geometric models with non-stationary spatial correlations in Lagrangian fluid flows, J. Nonlin. Sci., 28 (2018), 873-904.
doi: 10.1007/s00332-017-9431-0. |
[12] |
F. Gay-Balmaz and V. Putkaradze,
On noisy extensions of nonholonomic constraints, J. Nonlin. Sci., 26 (2016), 1571-1613.
doi: 10.1007/s00332-016-9313-x. |
[13] |
D. D. Holm, Variational principles for stochastic fluid dynamics, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 471 (2015), 20140963, 19 pp.
doi: 10.1098/rspa.2014.0963. |
[14] |
J. E. Marsden and A. Weinstein,
Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D: Nonlinear Phenomena, 7 (1983), 305-323.
doi: 10.1016/0167-2789(83)90134-3. |
[15] |
J. McWilliams,
A note on a consistent quasigeostrophic model in a multiply connected domain, Dynam. Atmos. Ocean, 1 (1977), 427-441.
doi: 10.1016/0377-0265(77)90002-1. |
[16] |
H. Yoshimura and F. Gay-Balmaz,
Hamilton–Pontryagin principle for incompressible ideal fluids, AIP Conference Proceedings, 1376 (2011), 645-647.
doi: 10.1063/1.3652002. |
[17] |
H. Yoshimura and J. E. Marsden,
Dirac structures in Lagrangian mechanics. I. Implicit Lagrangian systems, J. Geom. Phys., 57 (2006), 133-156.
doi: 10.1016/j.geomphys.2006.02.009. |
show all references
References:
[1] |
S. Albeverio, A. B. Cruzeiro and D. D. Holm, Stochastic Geometric Mechanics, Springer, 2017. |
[2] |
A. Arnaudon, A. L. de Castro and D. D. Holm,
Noise and dissipation on coadjoint orbits, J. Nonlin. Sci., 28 (2018), 91-145.
doi: 10.1007/s00332-017-9404-3. |
[3] |
V. I. Arnol'd,
Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Annales de l'institut Fourier, 16 (1966), 319-361.
doi: 10.5802/aif.233. |
[4] |
V. I. Arnol'd, Mathematical Methods of Classical Mechanics, volume 60 of Graduate Texts in Mathematics, Springer-Verlag, New York |
[5] |
J.-M. Bismut, Mécanique aléatoire, In Tenth Saint Flour Probability Summer School—1980 (Saint Flour, 1980), volume 929 of Lecture Notes in Math., pages 1–100, Springer, Berlin-New York, 1982. |
[6] |
N. Bou-Rabee and H. Owhadi,
Stochastic variational integrators, IMA J. Numer. Anal., 29 (2009), 421-443.
doi: 10.1093/imanum/drn018. |
[7] |
C. J. Cotter, G. A. Gottwald and D. D. Holm, Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics, Proc. Roy. Soc. A, 473 (2017), 20170388, 10pp.
doi: 10.1098/rspa.2017.0388. |
[8] |
C. J. Cotter, D. Crisan, D. D. Holm, W. Pan and I. Shevchenko, Modelling uncertainty using circulation-preserving stochastic transport noise in a 2-layer quasi-geostrophic model, arXiv preprint, arXiv: 1802.05711. |
[9] |
D. Crisan, F. Flandoli and D. Holm, Solution properties of a 3d stochastic Euler fluid equation, arXiv: 1704.06989, [math-ph], 2017.
doi: 10.1007/s00332-018-9506-6. |
[10] |
A. B. Cruzeiro, D. D. Holm and T. S. Ratiu,
Momentum maps and stochastic Clebsch action principles, Commun. in Math. Phys., 357 (2018), 873-912.
doi: 10.1007/s00220-017-3048-x. |
[11] |
F. Gay-Balmaz and D. D. Holm,
Stochastic geometric models with non-stationary spatial correlations in Lagrangian fluid flows, J. Nonlin. Sci., 28 (2018), 873-904.
doi: 10.1007/s00332-017-9431-0. |
[12] |
F. Gay-Balmaz and V. Putkaradze,
On noisy extensions of nonholonomic constraints, J. Nonlin. Sci., 26 (2016), 1571-1613.
doi: 10.1007/s00332-016-9313-x. |
[13] |
D. D. Holm, Variational principles for stochastic fluid dynamics, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 471 (2015), 20140963, 19 pp.
doi: 10.1098/rspa.2014.0963. |
[14] |
J. E. Marsden and A. Weinstein,
Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D: Nonlinear Phenomena, 7 (1983), 305-323.
doi: 10.1016/0167-2789(83)90134-3. |
[15] |
J. McWilliams,
A note on a consistent quasigeostrophic model in a multiply connected domain, Dynam. Atmos. Ocean, 1 (1977), 427-441.
doi: 10.1016/0377-0265(77)90002-1. |
[16] |
H. Yoshimura and F. Gay-Balmaz,
Hamilton–Pontryagin principle for incompressible ideal fluids, AIP Conference Proceedings, 1376 (2011), 645-647.
doi: 10.1063/1.3652002. |
[17] |
H. Yoshimura and J. E. Marsden,
Dirac structures in Lagrangian mechanics. I. Implicit Lagrangian systems, J. Geom. Phys., 57 (2006), 133-156.
doi: 10.1016/j.geomphys.2006.02.009. |
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