753 research outputs found
Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error
The ensemble Kalman filter and its variants have shown to be robust for data assimilation in high dimensional geophysical models, with localization, using ensembles of extremely small size relative to the model dimension. However, a reduced rank representation of the estimated covariance leaves a large dimensional complementary subspace unfiltered. Utilizing the dynamical properties of the filtration for the backward Lyapunov vectors, this paper explores a previously unexplained mechanism, providing a novel theoretical interpretation for the role of covariance inflation in ensemble-based Kalman filters. Our derivation of the forecast error evolution describes the dynamic upwelling of the unfiltered error from outside of the span of the anomalies into the filtered subspace. Analytical results for linear systems explicitly describe the mechanism for the upwelling, and the associated recursive Riccati equation for the forecast error, while nonlinear approximations are explored numerically
Analysis of the ensemble Kalman filter for inverse problems
The ensemble Kalman filter (EnKF) is a widely used methodology for state
estimation in partial, noisily observed dynamical systems, and for parameter
estimation in inverse problems. Despite its widespread use in the geophysical
sciences, and its gradual adoption in many other areas of application, analysis
of the method is in its infancy. Furthermore, much of the existing analysis
deals with the large ensemble limit, far from the regime in which the method is
typically used. The goal of this paper is to analyze the method when applied to
inverse problems with fixed ensemble size. A continuous-time limit is derived
and the long-time behavior of the resulting dynamical system is studied. Most
of the rigorous analysis is confined to the linear forward problem, where we
demonstrate that the continuous time limit of the EnKF corresponds to a set of
gradient flows for the data misfit in each ensemble member, coupled through a
common pre-conditioner which is the empirical covariance matrix of the
ensemble. Numerical results demonstrate that the conclusions of the analysis
extend beyond the linear inverse problem setting. Numerical experiments are
also given which demonstrate the benefits of various extensions of the basic
methodology
Affine Invariant Covariance Estimation for Heavy-Tailed Distributions
In this work we provide an estimator for the covariance matrix of a
heavy-tailed multivariate distributionWe prove that the proposed estimator
admits an \textit{affine-invariant} bound of the form
in high probability, where is the
unknown covariance matrix, and is the positive semidefinite
order on symmetric matrices. The result only requires the existence of
fourth-order moments, and allows for where is a measure of kurtosis of the
distribution, is the dimensionality of the space, is the sample size,
and is the desired confidence level. More generally, we can allow
for regularization with level , then gets replaced with the
degrees of freedom number. Denoting the condition
number of , the computational cost of the novel estimator is , which is comparable to the cost of the
sample covariance estimator in the statistically interesing regime .
We consider applications of our estimator to eigenvalue estimation with
relative error, and to ridge regression with heavy-tailed random design
An EnKF-Based Flow State Estimator for Aerodynamic Flows
Regardless of plant model, robust flow estimation based on limited measurements remains a major obstacle to successful flow control applications. Aiming to combine the robustness of a high-dimensional representation of the dynamics with the cost efficiency of a low-order approximation of the state covariance matrix, a flow state estimator based on the Ensemble Kalman Filter (EnKF) is applied to two-dimensional flow past a cylinder and an airfoil at high angle of attack and low Reynolds number. For the development purposes, we use the numerical algorithm as both the estimator and as a surrogate for the measurements. Estimation is successful using a reduced number of either pressure sensors on the surface of the body or sparsely placed velocity probes in the wake. Because the most relevant features of these flows is restricted to a low-dimensional subspace/manifold of the state space, asymptotic behavior of the estimator is shown to be achieved with a small ensemble size. The relative importance of each sensor location is evaluated by analyzing how they influence the estimated flow field. Covariance inflation is used to enhance the estimator performance in the presence of unmodeled free stream perturbations. A combination of parametric modeling and augmented state methodology is used to successfully estimate the forces on immersed bodies
Well Posedness and Convergence Analysis of the Ensemble Kalman Inversion
The ensemble Kalman inversion is widely used in practice to estimate unknown
parameters from noisy measurement data. Its low computational costs,
straightforward implementation, and non-intrusive nature makes the method
appealing in various areas of application. We present a complete analysis of
the ensemble Kalman inversion with perturbed observations for a fixed ensemble
size when applied to linear inverse problems. The well-posedness and
convergence results are based on the continuous time scaling limits of the
method. The resulting coupled system of stochastic differential equations
allows to derive estimates on the long-time behaviour and provides insights
into the convergence properties of the ensemble Kalman inversion. We view the
method as a derivative free optimization method for the least-squares misfit
functional, which opens up the perspective to use the method in various areas
of applications such as imaging, groundwater flow problems, biological problems
as well as in the context of the training of neural networks
Estimation of the signal subspace without estimation of the inverse covariance matrix
Let a high-dimensional random vector X can be represented as a sum of two components - a signal S, which belongs to some low-dimensional subspace S, and a noise component N. This paper presents a new approach for estimating the subspace S based on the ideas of the Non-Gaussian Component Analysis. Our approach avoids the technical difficulties that usually exist in similar methods - it doesn’t require neither the estimation of the inverse covariance matrix of X nor the estimation of the covariance matrix of N.dimension reduction, non-Gaussian components, NGCA
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