15,724 research outputs found
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
Guaranteed parameter estimation in nonlinear dynamic systems using improved bounding techniques
This paper is concerned with guaranteed parameter estimation in nonlinear dynamic systems in a context of bounded measurement error. The problem consists of finding - or approximating as closely as possible - the set of all possible parameter values such that the predicted outputs match the corresponding measurements within prescribed error bounds. An exhaustive search procedure is applied, whereby the parameter set is successively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a prespecified threshold on the approximation level is met. Exclusion tests rely on the ability to bound the solution set of the dynamic system for a given parameter subset and the tightness of these bounds is therefore paramount. Equally important is the time required to compute the bounds, thereby defining a trade-off. It is the objective of this paper to investigate this trade-off by comparing various bounding techniques based on interval arithmetic, Taylor model arithmetic and ellipsoidal calculus. When applied to a simple case study, ellipsoidal and Taylor model approaches are found to reduce the number of iterations significantly compared to interval analysis, yet the overall computational time is only reduced for tight approximation levels due to the computational overhead. © 2013 EUCA
Partial Observability and its Consistency for PDEs
In this paper, a quantitative measure of partial observability is defined for
PDEs. The quantity is proved to be consistent if the PDE is approximated using
well-posed approximation schemes. A first order approximation of an
unobservability index using an empirical Gramian is introduced. Several
examples are presented to illustrate the concept of partial observability,
including Burgers' equation and a one-dimensional nonlinear shallow water
equation.Comment: 5 figures, 25 pages. arXiv admin note: substantial text overlap with
arXiv:1111.584
The Consistency of Partial Observability for PDEs
In this paper, a new definition of observability is introduced for PDEs. It
is a quantitative measure of partial observability. The quantity is proved to
be consistent if approximated using well posed approximation schemes. A first
order approximation of an unobservability index using empirical gramian is
introduced. For linear systems with full state observability, the empirical
gramian is equivalent to the observability gramian in control theory. The
consistency of the defined observability is exemplified using a Burgers'
equation.Comment: 28 pages, 3 figure
Continuous-discrete unscented Kalman filtering framework by MATLAB ODE solvers and square-root methods
This paper addresses the problem of designing the {\it continuous-discrete}
unscented Kalman filter (UKF) implementation methods. More precisely, the aim
is to propose the MATLAB-based UKF algorithms for {\it accurate} and {\it
robust} state estimation of stochastic dynamic systems. The accuracy of the
{\it continuous-discrete} nonlinear filters heavily depends on how the
implementation method manages the discretization error arisen at the filter
prediction step. We suggest the elegant and accurate implementation framework
for tracking the hidden states by utilizing the MATLAB built-in numerical
integration schemes developed for solving ordinary differential equations
(ODEs). The accuracy is boosted by the discretization error control involved in
all MATLAB ODE solvers. This keeps the discretization error below the tolerance
value provided by users, automatically. Meanwhile, the robustness of the UKF
filtering methods is examined in terms of the stability to roundoff. In
contrast to the pseudo-square-root UKF implementations established in
engineering literature, which are based on the one-rank Cholesky updates, we
derive the stable square-root methods by utilizing the -orthogonal
transformations for calculating the Cholesky square-root factors
Approximate parameter inference in systems biology using gradient matching: a comparative evaluation
Background: A challenging problem in current systems biology is that of
parameter inference in biological pathways expressed as coupled ordinary
differential equations (ODEs). Conventional methods that repeatedly numerically
solve the ODEs have large associated computational costs. Aimed at reducing this
cost, new concepts using gradient matching have been proposed, which bypass
the need for numerical integration. This paper presents a recently established
adaptive gradient matching approach, using Gaussian processes, combined with a
parallel tempering scheme, and conducts a comparative evaluation with current
state of the art methods used for parameter inference in ODEs. Among these
contemporary methods is a technique based on reproducing kernel Hilbert spaces
(RKHS). This has previously shown promising results for parameter estimation,
but under lax experimental settings. We look at a range of scenarios to test the
robustness of this method. We also change the approach of inferring the penalty
parameter from AIC to cross validation to improve the stability of the method.
Methodology: Methodology for the recently proposed adaptive gradient
matching method using Gaussian processes, upon which we build our new
method, is provided. Details of a competing method using reproducing kernel
Hilbert spaces are also described here.
Results: We conduct a comparative analysis for the methods described in this
paper, using two benchmark ODE systems. The analyses are repeated under
different experimental settings, to observe the sensitivity of the techniques.
Conclusions: Our study reveals that for known noise variance, our proposed
method based on Gaussian processes and parallel tempering achieves overall the
best performance. When the noise variance is unknown, the RKHS method
proves to be more robust
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