The process of integrating observations into a numerical model of an evolving dynamical system, known as data assimilation, has become an essential tool in computational science. These methods, however, are computationally expensive as they typically involve large matrix multiplication and inversion. Furthermore, it is challenging to incorporate a constraint into the procedure, such as requiring a positive state vector. Here we introduce an entirely new approach to data assimilation, one that satisfies an information measure and uses the unnormalized Kullback-Leibler divergence, rather than the standard choice of Euclidean distance. Two sequential data assimilation algorithms are presented within this framework and are demonstrated numerically. These new methods are solved iteratively and do not require an adjoint. We find them to be computationally more efficient than Optimal Interpolation (3D-Var solution) and the Kalman filter whilst maintaining similar accuracy. Furthermore, these Kullback-Leibler data assimilation (KL-DA) methods naturally embed constraints, unlike Kalman filter approaches. They are ideally suited to systems that require positive valued solutions as the KL-DA guarantees this without need of transformations, projections, or any additional steps. This Kullback-Leibler framework presents an interesting new direction of development in data assimilation theory. The new techniques introduced here could be developed further and may hold potential for applications in the many disciplines that utilize data assimilation, especially where there is a need to evolve variables of large-scale systems that must obey physical constraints.
An interest is often present in knowing evolving variables that are not directly observable; this is the case in aerospace, engineering control, medical imaging, or data assimilation. What is at hand, though, are time-varying measured data, a model connecting them to variables of interest, and a model of how to evolve the variables over time. However, both models are only approximation and the observed data are tainted with noise. This is an ill-posed inverse problem. Methods, such as Kalman filter (KF), have been devised to extract the time-varying quantities of interest. These methods applied to this inverse problem, nonetheless, are slow, computation wise, since they require large matrices multiplications and even matrix inversion. Furthermore, these methods are not usually suitable to impose some constraints. This article introduces a new iterative filtering algorithm based on alternating projections. Experiments were run with simulated moving projectiles and were compared with results using KF. The new optimization algorithm proves to be slightly more accurate than KF, but, more to the point, it is much faster in terms of CPU time.
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