Multilevel ensemble Kalman filtering

Hoel, Håkon; Law, Kody J. H.; Tempone, Raúl

doi:10.1137/15m100955x

Cited by 90 publications

(124 citation statements)

References 46 publications

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“…The natural and yet challenging extension of the multilevel Monte Carlo (MLMC) framework to inference problems has recently been pioneered by the works [16,20,3,17], but, to the best knowledge of the authors, rigorous results for consistent filtering, via the particle filter, have yet to be obtained. In this article, the context of a partially observed diffusion is considered, with observations in discrete time; this will be detailed explicitly in the next section.…”

Section: Introductionmentioning

confidence: 99%

Multilevel Particle Filters

Jasra¹,

Kamatani²,

Law³

et al. 2017

SIAM J. Numer. Anal.

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176

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Abstract. In this paper the filtering of partially observed diffusions, with discrete-time observations, is considered. It is assumed that only biased approximations of the diffusion can be obtained, for choice of an accuracy parameter indexed by l. A multilevel estimator is proposed, consisting of a telescopic sum of increment estimators associated to the successive levels. The work associated to O(ε 2 ) mean-square error between the multilevel estimator and average with respect to the filtering distribution is shown to scale optimally, for example as O(ε −2 ) for optimal rates of convergence of the underlying diffusion approximation. The method is illustrated on some toy examples as well as estimation of interest rate based on real S&P 500 stock price data.

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Section: Introductionmentioning

confidence: 99%

Multilevel Particle Filters

Jasra¹,

Kamatani²,

Law³

et al. 2017

SIAM J. Numer. Anal.

Self Cite

176

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“…The motivation is typically the application to Bayesian inference problems or to sequential inference problems arising in the context of data assimilation and filtering. This includes the Metropolis-Hastings type Markov chain Monte Carlo estimators considered in this work, as well as importance sampling based multilevel Markov chain Monte Carlo methods [42], multilevel sequential Monte Carlo methods and multilevel particle filters [35,3,45,20,49,46], multilevel ensemble Kalman filters [44], multilevel approximate Bayesian computation [61], multilevel stochastic gradient Markov chain Monte Carlo algorithms [32] and multilevel methods based on ratio estimators [54,21]. We would also like to point out here that the "levels" in the references above are not necessarily defined through a hierarchy of meshes, but can also involve different numbers of particles (e.g.…”

mentioning

confidence: 99%

Multilevel Markov Chain Monte Carlo

Dodwell¹,

Ketelsen²,

Scheichl

et al. 2019

SIAM Rev.

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In this paper we address the problem of the prohibitively large computational cost of existing Markov chain Monte Carlo methods for large-scale applications with high-dimensional parameter spaces, e.g., in uncertainty quantification in porous media flow. We propose a new multilevel Metropolis-Hastings algorithm and give an abstract, problem-dependent theorem on the cost of the new multilevel estimator based on a set of simple, verifiable assumptions. For a typical model problem in subsurface flow, we then provide a detailed analysis of these assumptions and show significant gains over the standard Metropolis-Hastings estimator. Numerical experiments confirm the analysis and demonstrate the effectiveness of the method with consistent reductions of more than an order of magnitude in the cost of the multilevel estimator over the standard Metropolis-Hastings algorithm for tolerances ε < 10 −2 .

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“…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.…”

mentioning

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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Multilevel ensemble Kalman filtering

Cited by 90 publications

References 46 publications

Multilevel Particle Filters

Multilevel Particle Filters

Multilevel Markov Chain Monte Carlo

Multilevel Ensemble Transform Particle Filtering

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