Boujemaa Ait-El-Fquih

Efficient Kernel-Based Ensemble Gaussian Mixture Filtering

AbstractThe Bayesian filtering problem for data assimilation is considered following the kernel-based ensemble Gaussian mixture filtering (EnGMF) approach introduced by Anderson and Anderson. In this approach, the posterior distribution of the system state is propagated with the model using the ensemble Monte Carlo method, providing a forecast ensemble that is then used to construct a prior Gaussian mixture (GM) based on the kernel density estimator. This results in two update steps: a Kalman filter (KF)-like update of the ensemble members and a particle filter (PF)-like update of the weights, followed by a resampling step to start a new forecast cycle. After formulating EnGMF for any observational operator, the influence of the bandwidth parameter of the kernel function on the covariance of the posterior distribution is analyzed. Then the focus is on two aspects: (i) the efficient implementation of EnGMF with (relatively) small ensembles, where a new deterministic resampling strategy is proposed preservi...

Fast Kalman-Like Filtering for Large-Dimensional Linear and Gaussian State-Space Models

This article considers the filtering problem for linear and Gaussian state-space models with large dimensions, a setup in which the optimal Kalman Filter (KF) might not be applicable owing to the excessive cost of manipulating huge covariance matrices. Among the most popular alternatives that enable cheaper and reasonable computation is the Ensemble KF (EnKF), a Monte Carlo-based approximation. In this article, we consider a class of a posteriori distributions with diagonal covariance matrices and propose fast approximate deterministic-based algorithms based on the Variational Bayesian (VB) approach. More specifically, we derive two iterative KF-like algorithms that differ in the way they operate between two successive filtering estimates; one involves a smoothing estimate and the other involves a prediction estimate. Despite its iterative nature, the prediction-based algorithm provides a computational cost that is, on the one hand, independent of the number of iterations in the limit of very large state dimensions, and on the other hand, always much smaller than the cost of the EnKF. The cost of the smoothing-based algorithm depends on the number of iterations that may, in some situations, make this algorithm slower than the EnKF. The performances of the proposed filters are studied and compared to those of the KF and EnKF through a numerical example.

A Variational Bayesian Multiple Particle Filtering Scheme for Large-Dimensional Systems

This paper considers the Bayesian filtering problem in high-dimensional nonlinear state-space systems. In such systems, classical particle filters (PFs) are impractical due to the prohibitive number of required particles to obtain reasonable performances. One approach that has been introduced to overcome this problem is the concept of multiple PFs (MPFs), where the state-space is split into low-dimensional subspaces and then a separate PF is applied to each subspace. Remarkable performances of MPF-like filters motivated our investigation here into a new strategy that combines the variational Bayesian approach to split the state-space with random sampling techniques, to derive a new computationally efficient MPF. The propagation of each particle in the prediction step of the resulting filter requires generating only a single particle in contrast with standard MPFs, for which a set of (children) particles is required. We present simulation results to evaluate the behavior of the proposed filter and compare its performances against standard PF and a MPF.

An efficient multiple particle filter based on the variational Bayesian approach

This paper addresses the filtering problem in large-dimensional systems, in which conventional particle filters (PFs) remain computationally prohibitive owing to the large number of particles needed to obtain reasonable performances. To overcome this drawback, a class of multiple particle filters (MPFs) has been recently introduced in which the state-space is split into low-dimensional subspaces, and then a separate PF is applied to each subspace. In this paper, we adopt the variational Bayesian (VB) approach to propose a new MPF, the VBMPF. The proposed filter is computationally more efficient since the propagation of each particle requires generating one (new) particle only, while in the standard MPFs a set of (children) particles needs to be generated. In a numerical test, the proposed VBMPF behaves better than the PF and MPF.

Ensemble Kalman filtering with one-step-ahead smoothing

AbstractThe ensemble Kalman filter (EnKF) is widely used for sequential data assimilation. It operates as a succession of forecast and analysis steps. In realistic large-scale applications, EnKFs a...

An Efficient State-Parameter Filtering Scheme Combining Ensemble Kalman and Particle Filters

AbstractThis work addresses the state–parameter filtering problem for dynamical systems with relatively large-dimensional state and low-dimensional parameters’ vector. A Bayesian filtering algorithm combining the strengths of the particle filter (PF) and the ensemble Kalman filter (EnKF) is proposed. At each assimilation cycle of the proposed EnKF–PF, the PF is first used to sample the parameters’ ensemble followed by the EnKF to compute the state ensemble conditional on the resulting parameters’ ensemble. The proposed scheme is expected to be more efficient than the traditional state augmentation techniques, which suffer from the curse of dimensionality and inconsistency that is particularly pronounced when the state is a strongly nonlinear function of the parameters. In the new scheme, the EnKF and PF interact via their ensembles’ members, in contrast with the recently introduced two-stage EnKF–PF (TS–EnKF–PF), which exchanges point estimates between EnKF and PF while requiring almost double the computa...

A One-Step-Ahead Smoothing-Based Joint Ensemble Kalman Filter for State-Parameter Estimation of Hydrological Models

The ensemble Kalman filter (EnKF) recursively integrates field data into simulation models to obtain a better characterization of the model’s state and parameters. These are generally estimated following a state-parameters joint augmentation strategy. In this study, we introduce a new smoothing-based joint EnKF scheme, in which we introduce a one-step-ahead smoothing of the state before updating the parameters. Numerical experiments are performed with a two-dimensional synthetic subsurface contaminant transport model. The improved performance of the proposed joint EnKF scheme compared to the standard joint EnKF compensates for the modest increase in the computational cost.