Multilevel Particle Filters

Jasra, Ajay; Kamatani, Kengo; Law, Kody J. H.; Zhou, Yan

doi:10.1137/17m1111553

Cited by 88 publications

(178 citation statements)

References 26 publications

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“…The ETPF, coupled with localization, allows one to simply and cheaply carry out a multilevel coupling between each fine and coarse ensemble in each independent Monte Carlo estimator in the MLMC framework. A recent study has also proposed a framework to apply MLMC to nonlinear filtering with a modified random resampling step in the standard particle filtering methodology to couple particles from coarse and fine levels [12]. In contrast, the coupling in the present paper is designed to minimize the Wasserstein distance between the distributions of these transformed ensembles (in the standard ETPF methodology), originally suggested in [8].…”

Section: Discussionmentioning

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: Discussionmentioning

confidence: 99%

Multilevel Ensemble Transform Particle Filtering

Gregory¹,

Cotter²,

Reich³

2016

SIAM J. Sci. Comput.

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“…Sincev N n ,ỹ n ∈ ∩ r≥2 L r (Ω), inequalities (26) and (25) imply that v N,P n −v N n p (|I −K N n H| 2 + ỹ n − Hv N n 2p )P −1/2 + P −1 P −1/2 . The argument holds for any p ≥ 2, and the proof follows by induction.…”

Section: 1mentioning

confidence: 99%

“…These works were also the first to present L p convergence rates for weak approximations of MLEnKF in the large ensemble and finer numerical resolution limit. See [25,2,36,15,14,32,11] for recent contributions on multilevel methods in filtering.…”

Section: Introductionmentioning

confidence: 99%

Multilevel ensemble Kalman filtering

Hoel

Law

Tempone

2016

SIAM J. Numer. Anal.

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119

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Abstract. This work embeds a multilevel Monte Carlo (MLMC) sampling strategy into the Monte Carlo step of the ensemble Kalman filter (EnKF) in the setting of finite dimensional signal evolution and noisy discrete-time observations. The signal dynamics is assumed to be governed by a stochastic differential equation (SDE), and a hierarchy of time grids is introduced for multilevel numerical integration of that SDE. The resulting multilevel ensemble Kalman filter method (MLEnKF) is proved to asymptotically outperform EnKF in terms of computational cost vs. approximation accuracy. The theoretical results are illustrated numerically.

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“…It is also important to note that the pseudo-marginal approach is equally valid for Bayesian sampling strategies based on sequential Monte Carlo (Del Moral et al, 2006;Li et al, 2019;Sisson et al, 2007). Furthermore, advances in stochastic simulation (Schnoerr et al, 2017;Warne et al, 2019) can also improve the performance of the likelihood estimator, and the application of multilevel Monte Carlo to particle filters can further reduce estimator variance (Gregory et al, 2016;Jasra et al, 2017Jasra et al, , 2018.…”

Section: Discussionmentioning

confidence: 99%

A practical guide to pseudo-marginal methods for computational inference in systems biology

Warne

Baker

Simpson

2020

Journal of Theoretical Biology

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For many stochastic models of interest in systems biology, such as those describing biochemical reaction networks, exact quantification of parameter uncertainty through statistical inference is intractable. Likelihood-free computational inference techniques enable parameter inference when the likelihood function for the model is intractable but the generation of many sample paths is feasible through stochastic simulation of the forward problem. The most common likelihood-free method in systems biology is approximate Bayesian computation that accepts parameters that result in low discrepancy between stochastic simulations and measured data. However, it can be difficult to assess how the accuracy of the resulting inferences are affected by the choice of acceptance threshold and discrepancy function. The pseudo-marginal approach is an alternative likelihood-free inference method that utilises a Monte Carlo estimate of the likelihood function. This approach has several advantages, particularly in the context of noisy, partially observed, time-course data typical in biochemical reaction network studies. Specifically, the pseudo-marginal approach facilitates exact inference and uncertainty quantification, and may be efficiently combined with particle filters for low variance, high-accuracy likelihood estimation. In this review, we provide a practical introduction to the pseudo-marginal approach using inference for biochemical reaction networks as a series of case studies. Implementations of key algorithms and examples are provided using the Julia programming language; a high performance, open source programming language for scientific computing (https://github.com/davidwarne/Warne2019 GuideToPseudoMarginal).

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Multilevel Particle Filters

Cited by 88 publications

References 26 publications

Multilevel Ensemble Transform Particle Filtering

Multilevel Ensemble Transform Particle Filtering

Multilevel ensemble Kalman filtering

A practical guide to pseudo-marginal methods for computational inference in systems biology

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