Long-Time Asymptotics of the Filtering Distribution for Partially Observed Chaotic Dynamical Systems

Abstract

Abstract. The filtering distribution is a time-evolving probability distribution on the state of a dynamical system given noisy observations. We study the large-time asymptotics of this probability distribution for discrete-time, randomly initialized signals that evolve according to a deterministic map Ψ. The observations are assumed to comprise a low-dimensional projection of the signal, given by an operator P , subject to additive noise. We address the question of whether these observations contain sufficient information to accurately reconstruct the signal. In a general framework, we establish conditions on Ψ and P under which the filtering distributions concentrate around the signal in the small-noise, long-time asymptotic regime. Linear systems, the Lorenz '63 and '96 models, and the Navier-Stokes equation on a two-dimensional torus are within the scope of the theory. Our main findings come as a by-product of computable bounds, of independent interest, for suboptimal filters based on new variants of the 3DVAR filtering algorithm.

Almost Sure Error Bounds for Data Assimilation in Dissipative Systems with Unbounded Observation Noise

“…They show that for the 3DVAR scheme, the mean square of the error is of the same order of magnitude as the variance of the noise. In [21], Sanz-Alonso and Stuart extend this result, in expectation, to a wide class of dissipative PDEs, including infinite dimensional systems, that satisfy certain properties; the "absorbing ball" property and the "squeezing property". As is noted in [4], in a remark after Assumption 3.1, there is essentially a trade-off to be made between having bounded noise, with pointwise bounds, and unbounded noise, where similar techniques lead to results in expectation.…”

“…Other models typically studied in the context of data assimilation in geophysical applications (see e.g. [18,14,13,21]) are the Lorenz '63 and Lorenz '96 models, as they exhibit many of the properties of the N-S equations such as being dissipative with a quadratic and energy conserving nonlinearity, while having the advantage of being finite dimensional. Fortunately some remarkable properties of the 2D N-S equations have been known for some time.…”

Optimization Based Methods for Partially Observed Chaotic Systems

“…, v d ) ∈ R d , the solution of (1.1) will be denoted by Ψ t (v), or equivalently, v(t). Sanz-Alonso and Stuart [42] and Law et al [27] have assumed that the nonlinearity is energy conserving, i.e. B(v, v), v = 0 for every v ∈ R d .…”

“…(This is a commonly used value that is experimentally known to cause chaotic behaviour, see [32,33].) As shown on page 16 of Sanz-Alonso and Stuart [42], this system can be written in the form (1.1), and the bilinear form B(u, u) satisfies the energy-conserving property (i.e.…”

“…Equation (1.1) was shown in Sanz-Alonso and Stuart [42] and Law et al [27] to be applicable to three chaotic dynamical systems: the Lorenz 63' model, the Lorenz 96' model and the Navier-Stokes equation on the torus; such models have many applications. We note that instead of the trapping ball assumption, these papers have considered different assumptions on A and B(v, v).…”