We derive universal upper estimates for model prediction error under moderate but otherwise unknown model uncertainty. Our estimates give upper bounds on the leading-order trajectory uncertainty arising along model trajectories, solely as functions of the invariants of the known Cauchy–Green strain tensor of the model. Our bounds turn out to be optimal, which means that they cannot be improved for general systems. The quantity relating the leading-order trajectory-uncertainty to the model uncertainty is the model sensitivity (MS), which we find to be a useful tool for a quick global assessment of the impact of modeling uncertainties in various domains of the phase space. By examining the expectation that finite-time Lyapunov exponents capture sensitivity to modeling errors, we show that this does not generally follow. However, we find that certain important features of the finite-time Lyapunov exponent persist in the MS field.
REFERENCES
1.
R.
Kubo
, J. Phys. Soc. Jpn.
12
, 570
(1957
). 2.
D.
Ruelle
, Nonlinearity
22
, 855
(2009
). 3.
O.
Butterley
and C.
Liverani
, J. Mod. Dyn.
1
, 301
(2007
). 4.
M.
Hairer
and A. J.
Majda
, Nonlinearity
23
, 909
(2010
). 5.
R. V.
Abramov
and A. J.
Majda
, Nonlinearity
20
, 2793
(2007
). 6.
A. J.
Majda
and X.
Wang
, Commun. Math. Sci.
8
, 145
(2010
). 7.
V.
Lucarini
, J. Stat. Phys.
173
, 1698
(2018
). 8.
C. E.
Leith
, J. Atmos. Sci.
32
, 2022
(1975
). 9.
A.
Gritsun
and V.
Lucarini
, Physica D
349
, 62
(2017
). 10.
V.
Lembo
, V.
Lucarini
, and F.
Ragone
, Sci. Rep.
10
, 8668
(2020
). 11.
T.
Bódai
, V.
Lucarini
, and F.
Lunkeit
, Chaos
30
, 023124
(2020
). 12.
J. M.
Nese
, Physica D
35
, 237
(1989
). 13.
F.
Ginelli
, H.
Chaté
, R.
Livi
, and A.
Politi
, J. Phys. A Math. Theor.
46
, 254005
(2013
). 14.
E.
Hawkins
and R.
Sutton
, Bull. Am. Meteorol. Soc.
90
, 1095
(2009
). 15.
N.
Maher
, S.
Milinski
, L.
Suarez-Gutierrez
, M.
Botzet
, L.
Kornblueh
, Y.
Takano
, J.
Kröger
, R.
Ghosh
, C.
Hedemann
, C.
Li
et al., J. Adv. Model. Earth Syst.
11
, 2050
(2019
). 16.
T.
Tél
, T.
Bódai
, G.
Drótos
, T.
Haszpra
, M.
Herein
, B.
Kaszás
, and M.
Vincze
, J. Stat. Phys.
179
, 1496
(2020
).17.
E.
Ott
, B. R.
Hunt
, I.
Szunyogh
, A. V.
Zimin
, E. J.
Kostelich
, M.
Corazza
, E.
Kalnay
, D.
Patil
, and J. A.
Yorke
, Tellus Ser. A
56
, 415
(2004
). 18.
E.
Kalnay
, Atmospheric Modeling, Data Assimilation and Predictability
(Cambridge University Press
, 2002
).19.
C.
Grudzien
, A.
Carrassi
, and M.
Bocquet
, SIAM-ASA J. Uncertainty Quantif.
6
, 1335
(2018
). 20.
C.
Cardinali
, N.
Žagar
, G.
Radnoti
, and R.
Buizza
, Nonlinear Process. Geophys.
21
, 971
(2014
). 21.
C.
Piccolo
and M.
Cullen
, Mon. Weather Rev.
144
, 213
(2016
). 22.
M. A.
Laughton
, J. Electron. Control
17
, 577
(1964
). 23.
D. G.
Cacuci
, M.
Ionescu-Bujor
, and I. M.
Navon
, Sensitivity and Uncertainty Analysis, Volume II: Applications to Large-Scale Systems
(CRC Press
, 2005
).24.
D. J.
Lea
, M. R.
Allen
, and T. W.
Haine
, Tellus Ser. A
52
, 523
(2000
). 25.
D. J.
Lea
, T. W.
Haine
, M. R.
Allen
, and J. A.
Hansen
, Q. J. R. Meteorol. Soc.
128
, 2587
(2002
). 26.
J.
Thuburn
, Q. J. R. Meteorol. Soc.
131
, 73
(2005
). 27.
Q.
Wang
, J. Comput. Phys.
235
, 1
(2013
). 28.
Q.
Wang
, R.
Hu
, and P.
Blonigan
, J. Comput. Phys.
267
, 210
(2014
). 29.
M.
Ghil
and V.
Lucarini
, Rev. Mod. Phys.
92
, 035002
(2020
). 30.
A. Y.
Mitrophanov
, J. Appl. Probab.
40
, 970
(2003
). 31.
A. Y.
Mitrophanov
, Theory Probab. Appl.
50
, 319
(2006
). 32.
A.
Ni
and Q.
Wang
, J. Comput. Phys.
347
, 56
(2017
). 33.
D.
Lasagna
, A.
Sharma
, and J.
Meyers
, J. Comput. Phys.
391
, 119
(2019
). 34.
M.
Mathur
, G.
Haller
, T.
Peacock
, J. E.
Ruppert-Felsot
, and H. L.
Swinney
, Phys. Rev. Lett.
98
, 144502
(2007
). 35.
There exist various definitions of ridges in the literature,59,60 which are in general non-equivalent.61 Loosely speaking, we refer to locally maximizing hypersurfaces of the scalar field as ridges. To express this idea in more precise terms, we prefer to use the ridge definition in Ref. 55, where a ridge is defined to be a structurally stable, attracting invariant manifold of the gradient field with a structurally stable invariant boundary. This definition has the advantage that it is robust with respect to small perturbations of the scalar field. For the precise formulation, we refer to Definition 1.
36.
G.
Haller
, Annu. Rev. Fluid Mech.
47
, 137
(2015
). 37.
38.
G.
Haller
and T.
Sapsis
, Chaos
21
, 023115
(2011
). 39.
F.
Brauer
, J. Math. Anal. Appl.
14
, 198
(1966
). 40.
U.
Kirchgraber
, Celestial Mech.
14
, 351
(1976
). 41.
J.
Guckenheimer
and P.
Holmes
, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
(Springer-Verlag
, New York
, 1983
).42.
A.
Hadjighasem
, M.
Farazmand
, and G.
Haller
, Nonlinear Dyn.
73
, 689
(2013
). 43.
44.
Y. N.
Blagoveshchenskii
, Theory Probab. Appl.
7
, 115
(1962
). 45.
M. I.
Freidlin
and A. D.
Wentzell
, Random Perturbations of Dynamical Systems
(Springer
, 2012
).46.
47.
S.
Albeverio
and B.
Smii
, Stoch. Process. Their Appl.
125
, 1009
(2015
). 48.
M. S.
Gutiérrez
and V.
Lucarini
, J. Stat. Phys.
179
, 1572
(2020
). 49.
K.
Onu
, F.
Huhn
, and G.
Haller
, J. Comput. Sci.
7
, 26
(2015
). 50.
The mean-square of the trajectory uncertainty, , can be upper estimated by standard tools,43 using the Lipschitz property of and the bounds for the functions and . The estimate is of the form , which is a suitable form for applying Gronwall’s lemma.37 In our formulation, the bound can finally be expressed as , where is a Lipschitz constant for and and are the deterministic and stochastic modeling uncertainties, respectively.
51.
J. G.
Charney
and J. G.
DeVore
, J. Atmos. Sci.
36
, 1205
(1979
). 52.
D. T.
Crommelin
, J. D.
Opsteegh
, and F.
Verhulst
, J. Atmos. Sci.
61
, 1406
(2004
). 53.
H.
Babaee
, M.
Farazmand
, G.
Haller
, and T. P.
Sapsis
, Chaos
27
, 063103
(2017
). 54.
G.
Haller
, Physica D
240
, 574
(2011
). 55.
D.
Karrasch
and G.
Haller
, Chaos
23
, 043126
(2013
). 56.
N.
Fenichel
, Indiana Univ. Math. J.
21
, 193
(1971
). 57.
See https://github.com/balintkaszas/ModelSensitivity for access to the numerical implementation of the method.
58.
59.
D.
Eberly
, R.
Gardner
, B.
Morse
, S.
Pizer
, and C.
Scharlach
, J. Math. Imaging Vis.
4
, 353
(1994
). 60.
T.
Lindeberg
, Int. J. Comput. Vis.
30
, 117
(1998
). 61.
R.
Peikert
and F.
Sadlo
, in 2008 IEEE Pacific Visualization Symposium (IEEE, 2008), pp. 119–126.© 2020 Author(s).
2020
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