Frontiers of Mathematics in China >
Improved linear response for stochastically driven systems
Received date: 01 Feb 2011
Accepted date: 06 Jul 2011
Published date: 01 Apr 2012
Copyright
The recently developed short-time linear response algorithm, which predicts the average response of a nonlinear chaotic system with forcing and dissipation to small external perturbation, generally yields high precision of the response prediction, although suffers from numerical instability for long response times due to positive Lyapunov exponents. However, in the case of stochastically driven dynamics, one typically resorts to the classical fluctuationdissipation formula, which has the drawback of explicitly requiring the probability density of the statistical state together with its derivative for computation, which might not be available with sufficient precision in the case of complex dynamics (usually a Gaussian approximation is used). Here, we adapt the short-time linear response formula for stochastically driven dynamics, and observe that, for short and moderate response times before numerical instability develops, it is generally superior to the classical formula with Gaussian approximation for both the additive and multiplicative stochastic forcing. Additionally, a suitable blending with classical formula for longer response times eliminates numerical instability and provides an improved response prediction even for long response times.
Rafail V. ABRAMOV . Improved linear response for stochastically driven systems[J]. Frontiers of Mathematics in China, 2012 , 7(2) : 199 -216 . DOI: 10.1007/s11464-012-0192-7
1 |
Abramov R. Short-time linear response with reduced-rank tangent map. Chin Ann Math, 2009, 30B(5): 447-462
|
2 |
Abramov R. Approximate linear response for slow variables of dynamics with explicit time scale separation. J Comput Phys, 2010, 229(20): 7739-7746
|
3 |
Abramov R, Majda A. Blended response algorithms for linear fluctuation-dissipation for complex nonlinear dynamical systems. Nonlinearity, 2007, 20: 2793-2821
|
4 |
Abramov R, Majda A. New approximations and tests of linear fluctuation-response for chaotic nonlinear forced-dissipative dynamical systems. J Nonlin Sci, 2008, 18(3): 303-341
|
5 |
Abramov R, Majda A. New algorithms for low frequency climate response. J Atmos Sci, 2009, 66: 286-309
|
6 |
Bell T. Climate sensitivity from fluctuation dissipation: Some simple model tests. J Atmos Sci, 1980, 37(8): 1700-1708
|
7 |
Carnevale G, Falcioni M, Isola S, Purini R, Vulpiani A. Fluctuation-response in systems with chaotic behavior. Phys Fluids A, 1991, 3(9): 2247-2254
|
8 |
Cohen B, Craig G. The response time of a convective cloud ensemble to a change in forcing. Quart J Roy Met Soc, 2004, 130(598): 933-944
|
9 |
Eckmann J, Ruelle D. Ergodic theory of chaos and strange attractors. Rev Mod Phys, 1985, 57(3): 617-656
|
10 |
Evans D, Morriss G. Statistical Mechanics of Nonequilibrium Liquids. New York: Academic Press, 1990
|
11 |
Gritsun A. Fluctuation-dissipation theorem on attractors of atmospheric models. Russ J Numer Math Modeling, 2001, 16(2): 115-133
|
12 |
Gritsun A, Branstator G. Climate response using a three-dimensional operator based on the fluctuation-dissipation theorem. J Atmos Sci, 2007, 64: 2558-2575
|
13 |
Gritsun A, Branstator G, Dymnikov V. Construction of the linear response operator of an atmospheric general circulation model to small external forcing. Num Anal Math Modeling, 2002, 17: 399-416
|
14 |
Gritsun A, Branstator G, Majda A. Climate response of linear and quadratic functionals using the fluctuation dissipation theorem. J Atmos Sci, 2008, 65: 2824-2841
|
15 |
Gritsun A, Dymnikov V. Barotropic atmosphere response to small external actions, theory and numerical experiments. Atmos Ocean Phys, 1999, 35(5): 511-525
|
16 |
Kubo R, Toda M, Hashitsume N. Statistical Physics II: Nonequilibrium Statistical Mechanics. New York: Springer-Verlag, 1985
|
17 |
Kunita H. Stochastic Flows and Stochastic Differential Equations. Cambridge: Cambridge University Press, 1997
|
18 |
Leith C. Climate response and fluctuation-dissipation. J Atmos Sci, 1975, 32: 2022-2025
|
19 |
Lorenz E. Predictability—a problem partly solved. In: Palmer T, Hagedorn R, eds. Predictability of Weather and Climate. Cambridge: Cambridge University Press, 2006
|
20 |
Lorenz E, Emanuel K. Optimal sites for supplementary weather observations. J Atmos Sci, 1998, 55: 399-414
|
21 |
Majda A, Abramov R, Gershgorin B. High skill in low frequency climate response through fluctuation dissipation theorems despite structural instability. Proc Natl Acad Sci, 2010, 107(2): 581-586
|
22 |
Majda A, Abramov R, Grote M. Information Theory and Stochastics for Multiscale Nonlinear Systems. CRM Monograph Series of Centre de Recherches Mathématiques, Université de Montréal, Vol 25. Providence: American Mathematical Society, 2005
|
23 |
Majda A, Gershgorin B. A test model for fluctuation-dissipation theorems with timeperiodic statistics. Physica D, 2010, 239(17): 1741-1757
|
24 |
Majda A, Wang X. Linear response theory for statistical ensembles in complex systems with time-periodic forcing. Commun Math Sci, 2010, 8(1): 145-172
|
25 |
Risken F. The Fokker-Planck Equation. 2nd ed. New York: Springer-Verlag, 1989
|
26 |
Ruelle D. Chaotic Evolution and Strange Attractors. Cambridge: Cambridge University Press, 1989
|
27 |
Ruelle D. Differentiation of SRB states. Comm Math Phys, 1997, 187: 227-241
|
28 |
Ruelle D. General linear response formula in statistical mechanics, and the fluctuationdissipation theorem far from equilibrium. Phys Lett A, 1998, 245: 220-224
|
29 |
Young L-S. What are SRB measures, and which dynamical systems have them? J Stat Phys, 2002, 108(5-6): 733-754
|
/
〈 | 〉 |