6,912 research outputs found
Nonlinear stability of the ensemble Kalman filter with adaptive covariance inflation
The Ensemble Kalman filter and Ensemble square root filters are data
assimilation methods used to combine high dimensional nonlinear models with
observed data. These methods have proved to be indispensable tools in science
and engineering as they allow computationally cheap, low dimensional ensemble
state approximation for extremely high dimensional turbulent forecast models.
From a theoretical perspective, these methods are poorly understood, with the
exception of a recently established but still incomplete nonlinear stability
theory. Moreover, recent numerical and theoretical studies of catastrophic
filter divergence have indicated that stability is a genuine mathematical
concern and can not be taken for granted in implementation. In this article we
propose a simple modification of ensemble based methods which resolves these
stability issues entirely. The method involves a new type of adaptive
covariance inflation, which comes with minimal additional cost. We develop a
complete nonlinear stability theory for the adaptive method, yielding Lyapunov
functions and geometric ergodicity under weak assumptions. We present numerical
evidence which suggests the adaptive methods have improved accuracy over
standard methods and completely eliminate catastrophic filter divergence. This
enhanced stability allows for the use of extremely cheap, unstable forecast
integrators, which would otherwise lead to widespread filter malfunction.Comment: 34 pages. 4 figure
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.Comment: 38 page
Kalman-filter control schemes for fringe tracking. Development and application to VLTI/GRAVITY
The implementation of fringe tracking for optical interferometers is
inevitable when optimal exploitation of the instrumental capacities is desired.
Fringe tracking allows continuous fringe observation, considerably increasing
the sensitivity of the interferometric system. In addition to the correction of
atmospheric path-length differences, a decent control algorithm should correct
for disturbances introduced by instrumental vibrations, and deal with other
errors propagating in the optical trains. We attempt to construct control
schemes based on Kalman filters. Kalman filtering is an optimal data processing
algorithm for tracking and correcting a system on which observations are
performed. As a direct application, control schemes are designed for GRAVITY, a
future four-telescope near-infrared beam combiner for the Very Large Telescope
Interferometer (VLTI). We base our study on recent work in adaptive-optics
control. The technique is to describe perturbations of fringe phases in terms
of an a priori model. The model allows us to optimize the tracking of fringes,
in that it is adapted to the prevailing perturbations. Since the model is of a
parametric nature, a parameter identification needs to be included. Different
possibilities exist to generalize to the four-telescope fringe tracking that is
useful for GRAVITY. On the basis of a two-telescope Kalman-filtering control
algorithm, a set of two properly working control algorithms for four-telescope
fringe tracking is constructed. The control schemes are designed to take into
account flux problems and low-signal baselines. First simulations of the
fringe-tracking process indicate that the defined schemes meet the requirements
for GRAVITY and allow us to distinguish in performance. In a future paper, we
will compare the performances of classical fringe tracking to our Kalman-filter
control.Comment: 17 pages, 8 figures, accepted for publication in A&
Computable infinite dimensional filters with applications to discretized diffusion processes
Let us consider a pair signal-observation ((xn,yn),n 0) where the unobserved
signal (xn) is a Markov chain and the observed component is such that, given
the whole sequence (xn), the random variables (yn) are independent and the
conditional distribution of yn only depends on the corresponding state variable
xn. The main problems raised by these observations are the prediction and
filtering of (xn). We introduce sufficient conditions allowing to obtain
computable filters using mixtures of distributions. The filter system may be
finite or infinite dimensional. The method is applied to the case where the
signal xn = Xn is a discrete sampling of a one dimensional diffusion process:
Concrete models are proved to fit in our conditions. Moreover, for these
models, exact likelihood inference based on the observation (y0,...,yn) is
feasable
Norm minimized Scattering Data from Intensity Spectra
We apply the minimizing technique of compressive sensing (CS) to
non-linear quadratic observations. For the example of coherent X-ray scattering
we provide the formulae for a Kalman filter approach to quadratic CS and show
how to reconstruct the scattering data from their spatial intensity
distribution.Comment: 26 pages, 10 figures, reordered section
Kalman Filtering with Unknown Noise Covariances
Since it is often difficult to identify the noise covariances for a Kalman filter, they are commonly considered design variables. If so, we can as well try to choose them so that the corresponding Kalman filter has some nice form. In this paper, we introduce a one-parameter subfamily of Kalman filters with the property that the covariance parameters cancel in the expression for the Kalman gain. We provide a simple criterion which guarantees that the implicitly defined process covariance matrix is positive definite
- …