514 research outputs found
Modeling of flexible surfaces: A preliminary study
The one-dimensional classical scalar string equation which involves linearization about a horizontal reference or equilibrium position is derived. We then derive a model for small motion about a nonhorizontal reference. The implications of our findings to modeling of flexible antenna surfaces such as that in the Maypole Hoop/Column antenna are discussed
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
Moment bounds and geometric ergodicity of diffusions with random switching and unbounded transition rates
Improved linear response for stochastically driven systems
The recently developed short-time linear response algorithm, which predicts
the average response of a nonlinear chaotic system with forcing and dissipation
to small external perturbation, generally yields high precision of the response
prediction, although suffers from numerical instability for long response times
due to positive Lyapunov exponents. However, in the case of stochastically
driven dynamics, one typically resorts to the classical fluctuation-dissipation
formula, which has the drawback of explicitly requiring the probability density
of the statistical state together with its derivative for computation, which
might not be available with sufficient precision in the case of complex
dynamics (usually a Gaussian approximation is used). Here we adapt the
short-time linear response formula for stochastically driven dynamics, and
observe that, for short and moderate response times before numerical
instability develops, it is generally superior to the classical formula with
Gaussian approximation for both the additive and multiplicative stochastic
forcing. Additionally, a suitable blending with classical formula for longer
response times eliminates numerical instability and provides an improved
response prediction even for long response times
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