| Abstract: |
| We consider the problem of high-dimensional filtering/data assimilation of continuous and discrete state-space models at discrete times. This problem is particularly challenging as analytical solutions are usually not available and many numerical approximation methods can have a cost that scales exponentially with the dimension of the hidden state. We utilize a sequential Markov chain Monte Carlo method to obtain samples from an approximation of the filtering distribution. For certain state-space models, this method is proven to converge to the true filter as the number of samples, N, tends to infinity. We benchmark our algorithms on linear Gaussian state-space models against competing ensemble methods and demonstrate a significant improvement in both execution
speed and accuracy (the algorithm cost can range from O(Nd) to O(Nd[d+1]/2) based on the model noise covariance matrix structure, where d is the dimension of the hidden state. We then consider a state-space model with Lagrangian observations such that the spatial locations of these observations are unknown and
driven by the partially observed hidden signal. This problem is exceptionally challenging as not only is high-dimensional, but the model for the signal yields longer-range time dependencies through the observation locations. Finally, the algorithm is tested on the high-dimensional rotating shallow water model with real
data obtained from drifters in the ocean. |
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