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Peavoy, Daniel
Languages: English
Types: Doctoral thesis
Subjects: QA, QC
This thesis is about the construction of low dimensional diffusion models of climate\ud variables. It assesses the predictive skill of models derived from a principled averaging\ud procedure and a purely empirical approach. The averaging procedure starts from\ud the equations for the original system then approximates the \weather" variables by a\ud stochastic process. They are then averaged with respect to their invariant measure.\ud This assumes that they equilibriate much faster than the climate variables. The\ud empirical approach argues for a very general model form, then parameters are estimated\ud using likelihood based inference for Stochastic Differential Equations. This is\ud computationally demanding and relies upon Markov Chain Monte Carlo methods.\ud A large part of this thesis is focused upon techniques to improve the efficiency of\ud these algorithms.\ud The empirical approach works well on simple one dimensional models but\ud performs poorly on multivariate problems due to the rapid increase in unknown\ud parameters. The averaging procedure is skillful in multivariate problems but is\ud sensitive to lack of complete time scale separation in the system. In conclusion,\ud the averaging procedure is better and can be improved by estimating parameters in\ud a principled way based on the likelihood function and by including a latent noise\ud process in the model.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • 3.2 Invariant distributions for variables in Eq. (3.31) for = 0:1. . . . .
    • 3.3 Solution of the Galerkin truncation of the Burgers equation for times t = 0; 0:4; 1:5; 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
    • 3.4 Evolution of Fourier amplitudes for k = 1; 5; 10; 20 . . . . . . . . . .
    • 3.5 Example of dynamics of Mean ow Ut from Eq. (3.40) . . . . . . . .
    • 3.6 Comparison of predicted density from equilibrium statistical mechanics and the empirical density for the mean ow. . . . . . . . . . . . .
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