Xin T. Tong

Information barriers for noisy Lagrangian tracers in filtering random incompressible flows

An important practical problem is the recovery of a turbulent velocity field from Lagrangian tracers that move with the fluid flow. Here, the filtering skill of L moving Lagrangian tracers in recovering random incompressible flow fields defined through a finite number of random Fourier modes is studied with full mathematical rigour. Despite the inherent nonlinearity in measuring noisy Lagrangian tracers, it is shown below that there are exact closed analytic formulas for the optimal filter for the velocity field involving Riccati equations with random coefficients for the covariance matrix. This mathematical structure allows a detailed asymptotic analysis of filter performance, both as time goes to infinity and as the number of noisy Lagrangian tracers, L, increases. In particular, the asymptotic gain of information from L-tracers grows only like ln L in a precise fashion; i.e., an exponential increase in the number of tracers is needed to reduce the uncertainty by a fixed amount; in other words, there is a practical information barrier. The proofs proceed through a rigourous mean field approximation of the random Ricatti equation. Also, as an intermediate step, geometric ergodicity with respect to the uniform measure on the period domain is proved for any fixed number L of noisy Lagrangian tracers. All of the above claims are confirmed by detailed numerical experiments presented here.

Nonlinear stability and ergodicity of ensemble based Kalman filters

The ensemble Kalman filter (EnKF) and ensemble square root filter (ESRF) are data assimilation methods used to combine high dimensional, nonlinear dynamical models with observed data. Despite their widespread usage in climate science and oil reservoir simulation, very little is known about the long-time behavior of these methods and why they are effective when applied with modest ensemble sizes in large dimensional turbulent dynamical systems. By following the basic principles of energy dissipation and controllability of filters, this paper establishes a simple, systematic and rigorous framework for the nonlinear analysis of EnKF and ESRF with arbitrary ensemble size, focusing on the dynamical properties of boundedness and geometric ergodicity. The time uniform boundedness guarantees that the filter estimate will not diverge to machine infinity in finite time, which is a potential threat for EnKF and ESQF known as the catastrophic filter divergence. Geometric ergodicity ensures in addition that the filter has a unique invariant measure and that initialization errors will dissipate exponentially in time. We establish these results by introducing a natural notion of observable energy dissipation. The time uniform bound is achieved through a simple Lyapunov function argument, this result applies to systems with complete observations and strong kinetic energy dissipation, but also to concrete examples with incomplete observations. With the Lyapunov function argument established, the geometric ergodicity is obtained by verifying the controllability of the filter processes; in particular, such analysis for ESQF relies on a careful multivariate perturbation analysis of the covariance eigen-structure.

Concrete ensemble Kalman filters with rigorous catastrophic filter divergence

Significance Understanding ensemble-based data assimilation methods, including their performance when applied to high-dimensional nonlinear models with low ensemble size, is a crucial problem in science and engineering. Catastrophic filter divergence is a well-documented but mechanistically mysterious phenomenon whereby ensemble-state estimates explode to machine infinity despite the true state remaining in a bounded region. We provide breakthrough insight into the phenomenon by proposing a simple forecast model that experiences catastrophic filter divergence under all ensemble-based methods. This is the first instance to our knowledge of a forecast model that plainly and rigorously illustrates that simple mechanisms can lead to such a drastic filter malfunction and thereby sheds light on when catastrophic filter divergence should be expected and how it can be avoided. The ensemble Kalman filter and ensemble square root filters are data assimilation methods used to combine high-dimensional, nonlinear dynamical models with observed data. Ensemble methods are indispensable tools in science and engineering and have enjoyed great success in geophysical sciences, because they allow for computationally cheap low-ensemble-state approximation for extremely high-dimensional turbulent forecast models. From a theoretical perspective, the dynamical properties of these methods are poorly understood. One of the central mysteries is the numerical phenomenon known as catastrophic filter divergence, whereby ensemble-state estimates explode to machine infinity, despite the true state remaining in a bounded region. In this article we provide a breakthrough insight into the phenomenon, by introducing a simple and natural forecast model that transparently exhibits catastrophic filter divergence under all ensemble methods and a large set of initializations. For this model, catastrophic filter divergence is not an artifact of numerical instability, but rather a true dynamical property of the filter. The divergence is not only validated numerically but also proven rigorously. The model cleanly illustrates mechanisms that give rise to catastrophic divergence and confirms intuitive accounts of the phenomena given in past literature.

Noisy Lagrangian Tracers for Filtering Random Rotating Compressible Flows

The recovery of a random turbulent velocity field using Lagrangian tracers that move with the fluid flow is a practically important problem. This paper studies the filtering skill of

Intermittency in turbulent diffusion models with a mean gradient

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Rigorous Analysis for Efficient Statistically Accurate Algorithms for Solving Fokker--Planck Equations in Large Dimensions

L-noisy Lagrangian tracers in recovering random rotating compressible flows that are a linear combination of random incompressible geostrophically balanced (GB) flow and random rotating compressible gravity waves. The idealized random fields are defined through forced damped random amplitudes of Fourier eigenmodes of the rotating shallow-water equations with the rotation rate measured by the Rossby number

Nonlinear stability of the ensemble kalman filter with adaptive covariance inflation

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