Self-Similar Prior and Wavelet Bases for Hidden Incompressible Turbulent Motion

Abstract

Abstract. This work is concerned with the ill-posed inverse problem of estimating turbulent flows from the observation of an image sequence. From a Bayesian perspective, a divergence-free isotropic fractional Brownian motion (fBm) is chosen as a prior model for instantaneous turbulent velocity fields. This self-similar prior characterizes accurately second-order statistics of velocity fields in incompressible isotropic turbulence. Nevertheless, the associated maximum a posteriori involves a fractional Laplacian operator which is delicate to implement in practice. To deal with this issue, we propose to decompose the divergent-free fBm on well-chosen wavelet bases. As a first alternative, we propose to design wavelets as whitening filters. We show that these filters are fractional Laplacian wavelets composed with the Leray projector. As a second alternative, we use a divergence-free wavelet basis, which takes implicitly into account the incompressibility constraint arising from physics. Although the latter decomposition involves correlated wavelet coefficients, we are able to handle this dependence in practice. Based on these two wavelet decompositions, we finally provide effective and efficient algorithms to approach the maximum a posteriori. An intensive numerical evaluation proves the relevance of the proposed wavelet-based self-similar priors.

Reduced-order modeling of hidden dynamics

“…In the light of comments of Fig. 1, we believe that this error gap may be significantly reduced by using more relevant priors, such as those proposed in [16,17]. This is the topic of ongoing research.…”

“…These priors fulfill the domination condition. They are usually degenerated and defined by an inverse covariance q t ∈ R n×n of rank lower than n. The latter matrix implements typically some local regularity constraints, e.g., finite difference approximation of spatial gradients [26], or some self-similar constraints and long-range dependency proper to turbulent flows [16,17]. In this linear Gaussian setting, we obtain the posterior mean and covariance .…”

“…The definition of (2) usually stems from the inclusion of some knowledge on the physics and the nature of the uncertainties. Indeed, there often exist approximated probabilistic characterizations of deterministic chaotic systems, see e.g., [14,15,16,17] for turbulent systems.…”

“…Moreover, we assume that V t 's are zero-mean Gaussian random fields. Such priors are commonly used in computer vision [26], in fluid mechanics [16], or in geophysics [17] to describe the correlated structure of motion fields. These priors fulfill the domination condition.…”

Uncertainty of Atmospheric Motion Vectors by Sampling Tempered Posterior Distributions

“…In particular for fluid flows, standard choices for the covariance matrix correspond to smoothing the gradient of the divergence and vorticity of the flow [41]. More recent schemes structure long-range interactions of the displacement field using bivariate isotropic fractional Brownian motion (fBm) priors specified by two hyperparameters 2 [42,15]. Denoting the posterior by µ y and using Bayes' theorem, we get:…”