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Kadakia, N, Rey D, Ye J, Abarbanel HDI.  2017.  Symplectic structure of statistical variational data assimilation. Quarterly Journal of the Royal Meteorological Society. 143:756-771.   10.1002/qj.2962   Abstract

Data assimilation variational principles (4D-Var) exhibit a natural symplectic structure among the state variables x(t) and. x(t). We explore the implications of this structure in both Lagrangian coordinates {x(t), x(t)} andHamiltonian canonical coordinates {x(t), p(t)} through a numerical examination of the chaotic Lorenz 1996 model in ten dimensions. We find that there are a number of subtleties associated with discretization, boundary conditions, and symplecticity, suggesting differing approaches when working in the the Lagrangian versus the Hamiltonian description. We investigate these differences in detail, and accordingly develop a protocol for searching for optimal trajectories in a Hamiltonian space. We find that casting the problem into canonical coordinates can, in some situations, considerably improve the quality of predictions.

Schumann-Bischoff, J, Parlitz U, Abarbanel HDI, Kostuk M, Rey D, Eldridge M, Luther S.  2015.  Basin structure of optimization based state and parameter estimation. Chaos. 25   10.1063/1.4920942   AbstractWebsite

Most data based state and parameter estimation methods require suitable initial values or guesses to achieve convergence to the desired solution, which typically is a global minimum of some cost function. Unfortunately, however, other stable solutions (e.g., local minima) may exist and provide suboptimal or even wrong estimates. Here, we demonstrate for a 9-dimensional Lorenz-96 model how to characterize the basin size of the global minimum when applying some particular optimization based estimation algorithm. We compare three different strategies for generating suitable initial guesses, and we investigate the dependence of the solution on the given trajectory segment (underlying the measured time series). To address the question of how many state variables have to be measured for optimal performance, different types of multivariate time series are considered consisting of 1, 2, or 3 variables. Based on these time series, the local observability of state variables and parameters of the Lorenz-96 model is investigated and confirmed using delay coordinates. This result is in good agreement with the observation that correct state and parameter estimation results are obtained if the optimization algorithm is initialized with initial guesses close to the true solution. In contrast, initialization with other exact solutions of the model equations (different from the true solution used to generate the time series) typically fails, i.e., the optimization procedure ends up in local minima different from the true solution. Initialization using random values in a box around the attractor exhibits success rates depending on the number of observables and the available time series (trajectory segment). (C) 2015 AIP Publishing LLC.