Accuracy and stability of filters for dissipative PDEs

@article{Brett2013AccuracyAS,
  title={Accuracy and stability of filters for dissipative PDEs},
  author={Caroline Brett and Kei Fong Lam and Kody J. H. Law and David S. McCormick and M. R. Scott and Andrew M. Stuart},
  journal={Physica D: Nonlinear Phenomena},
  year={2013},
  volume={245},
  pages={34-45}
}

Figures from this paper

Almost Sure Error Bounds for Data Assimilation in Dissipative Systems with Unbounded Observation Noise
TLDR
The error of some simple data assimilation schemes in the presence of unbounded noise on a wide class of dissipative dynamical systems with certain properties is studied, including the Lorenz models and the 2D incompressible Navier-Stokes equations.
ACCURATE DATA ASSIMILATION FOR CHAOTIC DYNAMICAL SYSTEMS
TLDR
The key idea underlying the accuracy of the method can be summarized as follows: Unstable dynamical systems can be stabilized, and hence the solution recovered from noisy data, provided two conditions hold: observe enough of the system: the unstable modes.
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
Evaluating Data Assimilation Algorithms
TLDR
This paper examines the two-dimensional Navier–Stokes equations in a periodic geometry, which has these features and yet is tractable for explicit and accurate computation of the posterior distribution by state-of-the-art statistical sampling techniques.
Instability and Regularization for Data Assimilation
The process of blending observations and numerical models is called in the environmental sciences community, data assimilation. Data assimilation schemes produce an analysis state, which is the best
A Reduced Basis Ensemble Kalman Filter for State/parameter Identification in Large-scale Nonlinear Dynamical Systems
TLDR
This paper proposes a reduced basis ensemble Kalman filter technique to speed up the numerical solution of Bayesian inverse problems arising from the discretization of nonlinear time dependent PDEs.
Nonlinear error dynamics for cycled data assimilation methods
We investigate the error dynamics for cycled data assimilation systems, such that the inverse problem of state determination is solved at tk, k = 1, 2, 3, ?, with a first guess given by the state
Analysis of the 3DVAR filter for the partially observed Lorenz'63 model
TLDR
The goal of this paper is to analyze the 3DVAR method, using the Lorenz '63 model to exemplify the key ideas, and demonstrates how the widely used technique of variance inflation acts to stabilize the filter, and hence leads to asymptotic accuracy.
Downscaling data assimilation algorithm with applications to statistical solutions of the Navier–Stokes equations
...
1
2
3
4
...

References

SHOWING 1-10 OF 34 REFERENCES
Stability of Filters for the Navier-Stokes Equation
TLDR
Results are described showing that, in the small observational noise limit, the filters can be tuned to accurately track the signal itself (filter stability), provided the system is observed in a sufficiently large low dimensional space; roughly speaking this space should be large enough to contain the unstable modes of the linearized dynamics.
Accuracy and stability of the continuous-time 3DVAR filter for the Navier–Stokes equation
The 3DVAR filter is prototypical of methods used to combine observed data with a dynamical system, online, in order to improve estimation of the state of the system. Such methods are used for high
Nonlinear data assimilation in geosciences: an extremely efficient particle filter
Almost all research fields in geosciences use numerical models and observations and combine these using data-assimilation techniques. With ever-increasing resolution and complexity, the numerical
Evaluating Data Assimilation Algorithms
TLDR
This paper examines the two-dimensional Navier–Stokes equations in a periodic geometry, which has these features and yet is tractable for explicit and accurate computation of the posterior distribution by state-of-the-art statistical sampling techniques.
Filtering nonlinear dynamical systems with linear stochastic models
An important emerging scientific issue is the real time filtering through observations of noisy signals for nonlinear dynamical systems as well as the statistical accuracy of spatio-temporal
Data assimilation as a nonlinear dynamical systems problem: stability and convergence of the prediction-assimilation system.
TLDR
Stability of two nonlinear prediction-assimilation systems from dynamic meteorology is studied next via the complete spectrum of their Lyapunov exponents; these two systems are governed by a large set of ordinary and of partial differential equations, respectively.
Nonlinear error dynamics for cycled data assimilation methods
We investigate the error dynamics for cycled data assimilation systems, such that the inverse problem of state determination is solved at tk, k = 1, 2, 3, ?, with a first guess given by the state
Analysis methods for numerical weather prediction
TLDR
Methods discussed include variational techniques, smoothing splines, Kriging, optimal interpolation, successive corrections, constrained initialization, the Kalman-Bucy filter, and adjoint model data assimilation, which are all shown to relate to the idealized analysis, and hence to each other.
Obstacles to High-Dimensional Particle Filtering
Abstract Particle filters are ensemble-based assimilation schemes that, unlike the ensemble Kalman filter, employ a fully nonlinear and non-Gaussian analysis step to compute the probability
...
1
2
3
4
...