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Projected shadowing-based data assimilation. (English) Zbl 1406.37060

Summary: In this article we develop algorithms for data assimilation based upon a computational time dependent stable/unstable splitting. Our particular method is based upon shadowing refinement and synchronization techniques and is motivated by work on assimilation in the unstable subspace [A. Carrassi et al., Chaos 18, No. 2, 023112, 7 p. (2008; Zbl 1307.34064); Trevisan, D’Isidoro, and Talagrand, Q. J. R. Meteorol. Soc., 136 (2010), pp. 487–496; L. Palatella et al., J. Phys. A, Math. Theor. 46, No. 25, Article ID 254020, 19 p. (2013; Zbl 1351.37130)] and pseudo-orbit data assimilation [K. Judd et al., Physica D 151, No. 2–4, 125–141 (2001; Zbl 1052.62092); “ The geometry of model error”, J. Atmos. Sci. 65 1749–1772 (2008); H. Du and L. A. Smith, “Pseudo-orbit data assimilation. I: The perfect model scenario”, J. Atmos. Sci. 71, 469–482 (2014)]. The algorithm utilizes time dependent projections onto the nonstable subspace determined by employing computational techniques for Lyapunov exponents/vectors. The method is extended to parameter estimation without changing the problem dynamics and we address techniques for adapting the method when (as is commonly the case) observations are not available in the full model state space. We use a combination of analysis and numerical experiments (with the Lorenz 63 and Lorenz 96 models) to illustrate the efficacy of the techniques and show that the results compare favorably with other variational techniques.

MSC:

37M99 Approximation methods and numerical treatment of dynamical systems
37N99 Applications of dynamical systems
68P20 Information storage and retrieval of data
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