Can local particle filters beat the curse of dimensionality?

Rebeschini, Patrick; Handel, Ramon van

doi:10.1214/14-aap1061

Cited by 182 publications

(269 citation statements)

References 20 publications

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“…Standard sequential Monte Carlo (SMC) methods fail to track high dimensional systems due to exponentially degenerate importance weights. However, while there have only been some suggestions for a solution to this problem in SMCs, such as in [18], one can alter the above localization scheme in the ETPF to solve this problem swiftly [5]. It is also needed to reduce the computational cost of likelihood evaluations when the dimension of the state space is greater than the sample size…”

Section: The Multilevel Monte Carlo Methodmentioning

confidence: 99%

Multilevel Ensemble Transform Particle Filtering

Gregory¹,

Cotter²,

Reich³

2016

SIAM J. Sci. Comput.

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Abstract. This paper extends the multilevel Monte Carlo variance reduction technique to nonlinear filtering. In particular, multilevel Monte Carlo is applied to a certain variant of the particle filter, the ensemble transform particle filter (EPTF). A key aspect is the use of optimal transport methods to re-establish correlation between coarse and fine ensembles after resampling; this controls the variance of the estimator. Numerical examples present a proof of concept of the effectiveness of the proposed method, demonstrating significant computational cost reductions (relative to the single-level ETPF counterpart) in the propagation of ensembles.Key words. multilevel Monte Carlo, sequential data assimilation, optimal transport AMS subject classifications. 65C05, 62M20, 93E11, 93B40, 90C05 DOI. 10.1137/15M10382321. Introduction. Data assimilation is the process of incorporating observed data into model forecasts. In data assimilation, one is interested in computing statistics E η [X] of solutions X to random dynamical systems with respect to a posterior measure (η) given partial observations of the system. In particle filtering [7,4], this is done by using an empirical ensemble representing the posterior distribution η at any one time. The propagation in time of the members (particles) of this ensemble can be computationally expensive, especially in high dimensional systems.Recently, the multilevel Monte Carlo (MLMC) method has been developed for achieving significant cost reductions in Monte Carlo simulations [9]. It has been applied to areas such as Markov chain Monte Carlo [13] and quasi-Monte Carlo [10] to return computational cost reductions from existing techniques. It has also been applied to uncertainty quantification within PDEs [6]. The idea is to consider a hierarchy of discretized models, balancing numerical error in cheap/coarse models against Monte Carlo variance in expensive/fine models. It is desirable to adapt MLMC to sequential Monte Carlo methods such as particle filters, and some first steps have been taken in this direction. First, the authors of [11] have developed a multilevel ensemble Kalman filter (EnKF), using MLMC estimators to calculate the mean and covariance of the posterior, in the case where the underlying distributions are Gaussian and the model is linear. However, for non-Gaussian distributions and nonlinear models, the EnKF is biased. The method does, however, converge to a "mean-field limit" [14]. Second, the authors of [3] proposed a multilevel sequential Monte Carlo method for Bayesian inference problems to give significant computational cost reductions from standard techniques. Our goal in this paper is to take a step further by applying MLMC to nonlinear filtering problems.In general, MLMC works by computing statistics from pairs of coarser and finer

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Section: The Multilevel Monte Carlo Methodmentioning

confidence: 99%

Multilevel Ensemble Transform Particle Filtering

Gregory¹,

Cotter²,

Reich³

2016

SIAM J. Sci. Comput.

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“…In the following examples, the proposed sequential Langevin and Hamiltonian based MCMC algorithms will be compared to three different variants of SMC-based algorithms: standard SIR algorithm, the block SIR [6] (with a block size of 4) and a Resample-Move algorithm, denoted by SIR-RMK, for which K MCMC moves with the mHMC kernel described in Section IV-B is applied on each particle (for x n -i.e. L = 1) after the resampling stage.…”

Section: Numerical Simulations: Large Spatial Sensor Networkmentioning

confidence: 99%

“…Secondly, we can cite the Block Sequential Importance Resampling (Block SIR) approach in which the underlying idea is to partition the state space into separate subspaces of small dimensions and run one SMC algorithm on each subspace [6], [9]- [11]. However, this strategy introduces in the final estimates a difficult to quantify bias which depends on the position along the split state vector elements.…”

mentioning

confidence: 99%

Langevin and Hamiltonian Based Sequential MCMC for Efficient Bayesian Filtering in High-Dimensional Spaces

Septier

Peters²

2016

IEEE J. Sel. Top. Signal Process.

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Nonlinear non-Gaussian state-space models arise in numerous applications in statistics and signal processing. In this context, one of the most successful and popular approximation techniques is the Sequential Monte Carlo (SMC) algorithm, also known as particle filtering. Nevertheless, this method tends to be inefficient when applied to high dimensional problems.In this paper, we focus on another class of sequential inference methods, namely the Sequential Markov Chain Monte Carlo (SMCMC) techniques, which represent a promising alternative to SMC methods. After providing a unifying framework for the class of SMCMC approaches, we propose novel efficient strategies based on the principle of Langevin diffusion andHamiltonian dynamics in order to cope with the increasing number of high-dimensional applications. Simulation results show that the proposed algorithms achieve significantly better performance compared to existing algorithms.

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“…These methods are a good-choice from a theoretical point of view. Unfortunately, in practice, PFs do suffer from a degeneracy problem [33] (Section 1.4) and even more, many challenges have to be overcome before they can be considered under operational DA scenarios [34,35]. For these reasons, PFs are not considered any further in this paper.…”

Section: Gaussian Mixture Models Based Filtersmentioning

confidence: 99%

A Robust Non-Gaussian Data Assimilation Method for Highly Non-Linear Models

2018

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Abstract:In this paper, we propose an efficient EnKF implementation for non-Gaussian data assimilation based on Gaussian Mixture Models and Markov-Chain-Monte-Carlo (MCMC) methods. The proposed method works as follows: based on an ensemble of model realizations, prior errors are estimated via a Gaussian Mixture density whose parameters are approximated by means of an Expectation Maximization method. Then, by using an iterative method, observation operators are linearized about current solutions and posterior modes are estimated via a MCMC implementation. The acceptance/rejection criterion is similar to that of the Metropolis-Hastings rule. Experimental tests are performed on the Lorenz 96 model. The results show that the proposed method can decrease prior errors by several order of magnitudes in a root-mean-square-error sense for nearly sparse or dense observational networks.

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Can local particle filters beat the curse of dimensionality?

Cited by 182 publications

References 20 publications

Multilevel Ensemble Transform Particle Filtering

Multilevel Ensemble Transform Particle Filtering

Langevin and Hamiltonian Based Sequential MCMC for Efficient Bayesian Filtering in High-Dimensional Spaces

A Robust Non-Gaussian Data Assimilation Method for Highly Non-Linear Models

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