Improved variational methods in statistical data assimilation

Abstract

Abstract. Data assimilation transfers information from an observed system to a physically based model system with state variables x(t). The observations are typically noisy, the model has errors, and the initial state x(t 0 ) is uncertain: the data assimilation is statistical. One can ask about expected values of functions G(X) on the path X = {x(t 0 ), . . ., x(t m )} of the model state through the observation window t n = {t 0 , . . ., t m }. The conditional (on the measurements) probability distribution P (X) = exp[−A 0 (X)] determines these expected values. Variational methods using saddle points of the "action" A 0 (X), known as 4DVar (Talagrand and Courtier, 1987;Evensen, 2009), are utilized for estimating G(X) . In a path integral formulation of statistical data assimilation, we consider variational approximations in a realization of the action where measurement errors and model errors are Gaussian. We (a) discuss an annealing method for locating the path X 0 giving a consistent minimum of the action A 0 (X 0 ), (b) consider the explicit role of the number of measurements at each t n in determining A 0 (X 0 ), and (c) identify a parameter regime for the scale of model errors, which allows X 0 to give a precise estimate of G(X 0 ) with computable, small higher-order corrections.

Precision annealing Monte Carlo methods for statistical data assimilation and machine learning

“…We now turn to the precision annealing method suggested in [29,30,50]. It is used here to facilitate the search for the global minimum of the action A(X) as we gradually increase the model precision parameter R f .…”

“…The Laplace-method evaluations of expected value integrals is discussed in [27][28][29][30]. They do not sample from π(X | Y) away from its maximum.…”

“…Precision Annealing arises from the idea that if we choose an initial path for a Monte Carlo or a variational optimization [27][28][29][30] at the global minimum for small R f as we slowly increase R f , we will stay in a region of phase space where our sampling will arrive at the smallest minimum of the action, for a variational calculation, or sample the neighborhood of the maximum of π(X, P), for a Monte Carlo search. We adopted the following annealing schedule for R f :…”

“…since the precision annealing procedure locates the paths that dominate π(X | Y) ∝ exp[−A(X)] [27][28][29][30].…”

“…The model error decreases over a large range of R f until the numerical stability of the evaluation of this term is reduced as small errors in x(m + 1) − f (x(m), ν) are magnified by large values of R f . As this result appears, the action for each of the N I paths at each β levels off, becoming essentially independent of R f , and matches the measurement error, as it must do for consistency [27][28][29][30]. Until this point, we have examined outcomes of the RP Precision Annealing estimation procedure.…”