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Shadowing Pseudo-Orbits and Gradient Descent Noise Reduction

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Abstract

Shadowing trajectories are one of the most powerful ideas of modern dynamical systems theory, providing a tool for proving some central theorems and a means to assess the relevance of models and numerically computed trajectories of chaotic systems. Shadowing has also been seen to have a role in state estimation and forecasting of nonlinear systems. Shadowing trajectories are guaranteed to exist in hyperbolic systems, but this is not true of nonhyperbolic systems, indeed it can be shown there are systems that cannot have long shadowing trajectories. In this paper we consider what might be called shadowing pseudo-orbits. These are pseudo-orbits that remain close to a given pseudo-orbit, but have smaller mismatches between forecast state and verifying state. Shadowing pseudo-orbits play a useful role in the understanding and analysis of gradient descent noise reduction, state estimation, and forecasting nonlinear systems, because their existence can be ensured for a wide class of nonhyperbolic systems. New theoretical results are presented that extend classical shadowing theorems to shadowing pseudo-orbits. These new results provide some insight into the convergence behaviour of gradient descent noise reduction methods. The paper also discusses, in the light of the new results, some recent numerical results for an operational weather forecasting model when gradient descent noise reduction was employed.

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References

  • Anosov, D.V.: Geodesic flows and closed Riemannian manifolds with negative curvature. Proc. Steklov Inst. Math. 90, 235 (1967)

    Google Scholar 

  • Baker, N.E., Daley, R., Gelaro, R., Goerss, J., Hogan, T., Langland, R., Pauley, R., Rennick, M., Reynolds, C., Rohaly, G., Rosmond, T., Swadley, S.: The Navy operational global atmospheric prediction system: A brief history of past, present, and future developments. Technical Report, Naval Research Laboratory, Monterey, California (1998)

  • Bowen, R.: ω-limit sets for axiom A diffeomorphisms. J. Differ. Equ. 18, 333–339 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  • Brocker, J., Parlitz, U., Ogorzalek, M.: Nonlinear noise reduction. Proc. IEEE 90(5), 898–918 (2002)

    Article  Google Scholar 

  • Chow, S.E., Palmer, K.J.: On the numerical computation of orbits of dynamical systems: The higher dimensional case. J. Complex. 8, 398–423 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  • Daley, R., Barker, E.: NAVDAS: Formulation and diagnostics. Mon. Weather Rev. 129, 869–883 (2001)

    Article  Google Scholar 

  • Davies, M.: Noise reduction by gradient descent. Int. J. Bifurc. Chaos 3(1), 113–118 (1992)

    Google Scholar 

  • Dawson, S., Grebogi, C., Sauer, T., Yorke, J.A.: Obstructions to shadowing when a lyapunov exponent fluctuates about zero. Phys. Rev. Lett. 73, 1927 (1994)

    Article  Google Scholar 

  • Diamond, P., Kloeden, P., Kozyakin, V., Pokrovskii, A.: Robustness of observed behaviour of semi-hyperbolic dynamical systems. Avtom. Telemek. 11 (1994)

  • Diamond, P., Kloeden, P., Kozyakin, V., Pokrovskii, A.: Semihyperbolic mappings. J. Nonlin. Sci. 5, 419–431 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  • Gilmour, I.: Nonlinear model evaluation: Iota-shadowing, probabilistic prediction and weather forecasting. Ph.D. Thesis, Mathematical Institute, Oxford University (1998)

  • Hammel, S.M., Yorke, J.A., Grebogi, C.: Do numerical orbits of chaotic dynamical processes represent true orbits? Complexity 3, 136–145 (1987a)

    Article  MATH  MathSciNet  Google Scholar 

  • Hammel, S.M., Yorke, J.A., Grebogi, C.: Numerical orbits of chaotic processes represent true orbits. Bull. (New Ser.) Am. Math. Soc. 19, 465 (1987b)

    MathSciNet  Google Scholar 

  • Judd, K.: Nonlinear state estimation, indistinguishable states and the extended Kalman filter. Physica D 183, 273–281 (2003)

    MATH  MathSciNet  Google Scholar 

  • Judd, K., Smith, L.A.: Indistinguishable states I: Perfect model scenario. Physica D 151, 125–141 (2001)

    MATH  MathSciNet  Google Scholar 

  • Judd, K., Smith, L.A.: Indistinguishable states II: Imperfect model scenarios. Physica D 196, 224–242 (2004)

    MATH  MathSciNet  Google Scholar 

  • Judd, K., Reynolds, C.A., Rosmond, T.: Toward shadowing in operational weather prediction. Technical Report NRL/MR/7530-04-18, Naval Research Laboratory, Monterey, CA 93943-5502, USA (2004a)

  • Judd, K., Smith, L.A., Weisheimer, A.: Gradient free descent: Shadowing and state estimation with limited derivative information. Physica D 190, 153–166 (2004b)

    MATH  MathSciNet  Google Scholar 

  • Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and Its Applications, vol. 54. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  • Kloeden, P.E., Ombach, J., Pokrovksii, A.V.: Continuous and inverse shadowing. J. Funct. Diff. Equ. 6, 135–151 (1999)

    Google Scholar 

  • Kostelich, E., Kan, I., Grebogi, C., Ott, E., Yorke, J.A.: Unstable dimension variability: A source of nonhyperbolicity in chaotic systems. Physica D 109, 81 (1997)

    MATH  MathSciNet  Google Scholar 

  • Lai, Y.C., Grebogi, C., Kruths, J.: Modeling of deterministic chaotic systems. Phys. Rev. E 59(3), 2907–2910 (1999)

    Google Scholar 

  • Livsic, A.: Certain properties of the homology of Y-systems. Mat. Zamet. 10, 555–564 (1971)

    MathSciNet  Google Scholar 

  • Nitica, V., Pollicott, M.: Transitivity of euclidean extensions of anosov diffeomorphisms. Ergod. Theory Dyn. Syst. 25(1), 257–270 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Orrell, D., Smith, L.A., Barkmeijer, J., Palmer, T.: Model error in weather forecasting. Nonlinear Process. Geophys. 8, 357–371 (2001)

    Google Scholar 

  • Pilyugin, S.Y., Plamenevskaya, O.B.: Shadowing is generic. Topol. Appl. 97, 253–266 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Pollicott, M.: Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds, vol. 180. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  • Ridout, D., Judd, K.: Convergence properties of gradient descent noise reduction. Physica D 165, 27–48 (2001)

    MathSciNet  Google Scholar 

  • Sauer, T., Grebogi, C., Yorke, J.: How long do numerical chaotic solutions remain valid? Phys. Rev. Lett. 79(1), 59–62 (1997)

    Article  Google Scholar 

  • Smith, L.A., Gilmour, I.: Accountability and internal consistency in ensemble formation. In: Proceedings of a Workshop on Predicability, vol. 133. ECMWF, Reading (1997)

    Google Scholar 

  • Viana, R.L., Barbosa, J.R.R., Grebogi, C., Batista, A.: Simulating a chaotic process. Braz. J. Phys. 35(1), 1–9 (2005)

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, 6009, WA, Australia

    Kevin Judd

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  1. Kevin Judd
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Correspondence to Kevin Judd.

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Communicated by K. Aihara.

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Judd, K. Shadowing Pseudo-Orbits and Gradient Descent Noise Reduction. J Nonlinear Sci 18, 57–74 (2008). https://doi.org/10.1007/s00332-007-9010-x

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  • DOI: https://doi.org/10.1007/s00332-007-9010-x

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