15,341 research outputs found
Second-order accurate ensemble transform particle filters
Particle filters (also called sequential Monte Carlo methods) are widely used for state and parameter estimation problems in the context of nonlinear evolution equations. The recently proposed ensemble transform particle filter (ETPF) (S.~Reich, {\it A non-parametric ensemble transform method for Bayesian inference}, SIAM J.~Sci.~Comput., 35, (2013), pp. A2013--A2014) replaces the resampling step of a standard particle filter by a linear transformation which allows for a hybridization of particle filters with ensemble Kalman filters and renders the resulting hybrid filters applicable to spatially extended systems. However, the linear transformation step is computationally expensive and leads to an underestimation of the ensemble spread for small and moderate ensemble sizes. Here we address both of these shortcomings by developing second-order accurate extensions of the ETPF. These extensions allow one in particular to replace the exact solution of a linear transport problem by its Sinkhorn approximation. It is also demonstrated that the nonlinear ensemble transform filter (NETF) arises as a special case of our general framework. We illustrate the performance of the second-order accurate filters for the chaotic Lorenz-63 and Lorenz-96 models and a dynamic scene-viewing model. The numerical results for the Lorenz-63 and Lorenz-96 models demonstrate that significant accuracy improvements can be achieved in comparison to a standard ensemble Kalman filter and the ETPF for small to moderate ensemble sizes. The numerical results for the scene-viewing model reveal, on the other hand, that second-order corrections can lead to statistically inconsistent samples from the posterior parameter distribution
Accounting for model error in Tempered Ensemble Transform Particle Filter and its application to non-additive model error
In this paper, we trivially extend Tempered (Localized) Ensemble Transform
Particle Filter---T(L)ETPF---to account for model error. We examine T(L)ETPF
performance for non-additive model error in a low-dimensional and a
high-dimensional test problem. The former one is a nonlinear toy model, where
uncertain parameters are non-Gaussian distributed but model error is Gaussian
distributed. The latter one is a steady-state single-phase Darcy flow model,
where uncertain parameters are Gaussian distributed but model error is
non-Gaussian distributed. The source of model error in the Darcy flow problem
is uncertain boundary conditions. We comapare T(L)ETPF to a Regularized
(Localized) Ensemble Kalman Filter---R(L)EnKF. We show that T(L)ETPF
outperforms R(L)EnKF for both the low-dimensional and the high-dimensional
problem. This holds even when ensemble size of TLETPF is 100 while ensemble
size of R(L)EnKF is greater than 6000. As a side note, we show that TLETPF
takes less iterations than TETPF, which decreases computational costs; while
RLEnKF takes more iterations than REnKF, which incerases computational costs.
This is due to an influence of localization on a tempering and a regularizing
parameter
Application of ensemble transform data assimilation methods for parameter estimation in reservoir modelling
Over the years data assimilation methods have been developed to obtain
estimations of uncertain model parameters by taking into account a few
observations of a model state. The most reliable methods of MCMC are
computationally expensive. Sequential ensemble methods such as ensemble Kalman
filers and particle filters provide a favourable alternative. However, Ensemble
Kalman Filter has an assumption of Gaussianity. Ensemble Transform Particle
Filter does not have this assumption and has proven to be highly beneficial for
an initial condition estimation and a small number of parameter estimation in
chaotic dynamical systems with non-Gaussian distributions. In this paper we
employ Ensemble Transform Particle Filter (ETPF) and Ensemble Transform Kalman
Filter (ETKF) for parameter estimation in nonlinear problems with 1, 5, and
2500 uncertain parameters and compare them to importance sampling (IS). We
prove that the updated parameters obtained by ETPF lie within the range of an
initial ensemble, which is not the case for ETKF. We examine the performance of
ETPF and ETKF in a twin experiment setup and observe that for a small number of
uncertain parameters (1 and 5) ETPF performs comparably to ETKF in terms of the
mean estimation. For a large number of uncertain parameters (2500) ETKF is
robust with respect to the initial ensemble while ETPF is sensitive due to
sampling error. Moreover, for the high-dimensional test problem ETPF gives an
increase in the root mean square error after data assimilation is performed.
This is resolved by applying distance-based localization, which however
deteriorates a posterior estimation of the leading mode by largely increasing
the variance. A possible remedy is instead of applying localization to use only
leading modes that are well estimated by ETPF, which demands a knowledge at
which mode to truncate
Data assimilation in the low noise regime with application to the Kuroshio
On-line data assimilation techniques such as ensemble Kalman filters and
particle filters lose accuracy dramatically when presented with an unlikely
observation. Such an observation may be caused by an unusually large
measurement error or reflect a rare fluctuation in the dynamics of the system.
Over a long enough span of time it becomes likely that one or several of these
events will occur. Often they are signatures of the most interesting features
of the underlying system and their prediction becomes the primary focus of the
data assimilation procedure. The Kuroshio or Black Current that runs along the
eastern coast of Japan is an example of such a system. It undergoes infrequent
but dramatic changes of state between a small meander during which the current
remains close to the coast of Japan, and a large meander during which it bulges
away from the coast. Because of the important role that the Kuroshio plays in
distributing heat and salinity in the surrounding region, prediction of these
transitions is of acute interest. Here we focus on a regime in which both the
stochastic forcing on the system and the observational noise are small. In this
setting large deviation theory can be used to understand why standard filtering
methods fail and guide the design of the more effective data assimilation
techniques. Motivated by our analysis we propose several data assimilation
strategies capable of efficiently handling rare events such as the transitions
of the Kuroshio. These techniques are tested on a model of the Kuroshio and
shown to perform much better than standard filtering methods.Comment: 43 pages, 12 figure
Deterministic Mean-field Ensemble Kalman Filtering
The proof of convergence of the standard ensemble Kalman filter (EnKF) from
Legland etal. (2011) is extended to non-Gaussian state space models. A
density-based deterministic approximation of the mean-field limit EnKF
(DMFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given
a certain minimal order of convergence between the two, this extends
to the deterministic filter approximation, which is therefore asymptotically
superior to standard EnKF when the dimension . The fidelity of
approximation of the true distribution is also established using an extension
of total variation metric to random measures. This is limited by a Gaussian
bias term arising from non-linearity/non-Gaussianity of the model, which exists
for both DMFEnKF and standard EnKF. Numerical results support and extend the
theory
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