2,318 research outputs found
Discretizing stochastic dynamical systems using Lyapunov equations
Stochastic dynamical systems are fundamental in state estimation, system
identification and control. System models are often provided in continuous
time, while a major part of the applied theory is developed for discrete-time
systems. Discretization of continuous-time models is hence fundamental. We
present a novel algorithm using a combination of Lyapunov equations and
analytical solutions, enabling efficient implementation in software. The
proposed method circumvents numerical problems exhibited by standard algorithms
in the literature. Both theoretical and simulation results are provided
A detectability criterion and data assimilation for non-linear differential equations
In this paper we propose a new sequential data assimilation method for
non-linear ordinary differential equations with compact state space. The method
is designed so that the Lyapunov exponents of the corresponding estimation
error dynamics are negative, i.e. the estimation error decays exponentially
fast. The latter is shown to be the case for generic regular flow maps if and
only if the observation matrix H satisfies detectability conditions: the rank
of H must be at least as great as the number of nonnegative Lyapunov exponents
of the underlying attractor. Numerical experiments illustrate the exponential
convergence of the method and the sharpness of the theory for the case of
Lorenz96 and Burgers equations with incomplete and noisy observations
Linear theory for filtering nonlinear multiscale systems with model error
We study filtering of multiscale dynamical systems with model error arising
from unresolved smaller scale processes. The analysis assumes continuous-time
noisy observations of all components of the slow variables alone. For a linear
model with Gaussian noise, we prove existence of a unique choice of parameters
in a linear reduced model for the slow variables. The linear theory extends to
to a non-Gaussian, nonlinear test problem, where we assume we know the optimal
stochastic parameterization and the correct observation model. We show that
when the parameterization is inappropriate, parameters chosen for good filter
performance may give poor equilibrium statistical estimates and vice versa.
Given the correct parameterization, it is imperative to estimate the parameters
simultaneously and to account for the nonlinear feedback of the stochastic
parameters into the reduced filter estimates. In numerical experiments on the
two-layer Lorenz-96 model, we find that parameters estimated online, as part of
a filtering procedure, produce accurate filtering and equilibrium statistical
prediction. In contrast, a linear regression based offline method, which fits
the parameters to a given training data set independently from the filter,
yields filter estimates which are worse than the observations or even divergent
when the slow variables are not fully observed
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
Mathematical control of complex systems
Copyright © 2013 ZidongWang et al.This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
Convergence of discrete time Kalman filter estimate to continuous time estimate
This article is concerned with the convergence of the state estimate obtained
from the discrete time Kalman filter to the continuous time estimate as the
temporal discretization is refined. We derive convergence rate estimates for
different systems, first finite dimensional and then infinite dimensional with
bounded or unbounded observation operators. Finally, we derive the convergence
rate in the case where the system dynamics is governed by an analytic
semigroup. The proofs are based on applying the discrete time Kalman filter on
a dense numerable subset of a certain time interval .Comment: Author's version of the manuscript accepted for publication in
International Journal of Contro
Well-Posedness And Accuracy Of The Ensemble Kalman Filter In Discrete And Continuous Time
The ensemble Kalman filter (EnKF) is a method for combining a dynamical model
with data in a sequential fashion. Despite its widespread use, there has been
little analysis of its theoretical properties. Many of the algorithmic
innovations associated with the filter, which are required to make a useable
algorithm in practice, are derived in an ad hoc fashion. The aim of this paper
is to initiate the development of a systematic analysis of the EnKF, in
particular to do so in the small ensemble size limit. The perspective is to
view the method as a state estimator, and not as an algorithm which
approximates the true filtering distribution. The perturbed observation version
of the algorithm is studied, without and with variance inflation. Without
variance inflation well-posedness of the filter is established; with variance
inflation accuracy of the filter, with resepct to the true signal underlying
the data, is established. The algorithm is considered in discrete time, and
also for a continuous time limit arising when observations are frequent and
subject to large noise. The underlying dynamical model, and assumptions about
it, is sufficiently general to include the Lorenz '63 and '96 models, together
with the incompressible Navier-Stokes equation on a two-dimensional torus. The
analysis is limited to the case of complete observation of the signal with
additive white noise. Numerical results are presented for the Navier-Stokes
equation on a two-dimensional torus for both complete and partial observations
of the signal with additive white noise
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