Particle Filtering for Stochastic Navier--Stokes Signal Observed with Linear Additive Noise

Llopis, Francesc Pons; Kantas, Nikolas; Beskos, Alexandros; Jasra, Ajay

doi:10.1137/17m1151900

Cited by 25 publications

(17 citation statements)

References 45 publications

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“…In addition, if there is a well‐defined limit of the model (in some sense) as

d

grows, useful particle filters can be developed (see e.g. Kantas et al , 2014; Llopis et al , 2018). As noted in Chatterjee and Diaconis (2018), the key criterion that needs to be satisfied is that the target distribution,

\frac{{falsefalse}_{false\prod}^{i = 1} G (u_{i}, y_{i}) Q^{L} (u_{(i - 1)}, u_{i})}{\int {falsefalse}_{false\prod}^{i = 1} G (u_{i}, y_{i}) Q^{L} (u_{(i - 1)}, u_{i}) d u_{1 : k}},

and the importance distribution,

q_{1} (u_{1}) {falsefalse}_{\prod}^{i = 2} q_{i} (u_{i - 1}, u_{i}),

do not become mutually singular in the limit

d \to \infty

(or equivalently that the symmetrized Kullback–Liebler distance between these distributions does not explode with

d

).…”

Section: Some Computational Methodsmentioning

confidence: 99%

Advanced Multilevel Monte Carlo Methods

Jasra

Law

Suciu

2020

Int Statistical Rev

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This article reviews the application of advanced Monte Carlo techniques in the context of Multilevel Monte Carlo (MLMC). MLMC is a strategy employed to compute expectations which can be biased in some sense, for instance, by using the discretization of a associated probability law. The MLMC approach works with a hierarchy of biased approximations which become progressively more accurate and more expensive. Using a telescoping representation of the most accurate approximation, the method is able to reduce the computational cost for a given level of error versus i.i.d. sampling from this latter approximation. All of these ideas originated for cases where exact sampling from couples in the hierarchy is possible. This article considers the case where such exact sampling is not currently possible. We consider Markov chain Monte Carlo and sequential Monte Carlo methods which have been introduced in the literature and we describe different strategies which facilitate the application of MLMC within these methods.

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“…In addition, if there is a well‐defined limit of the model (in some sense) as

d

\frac{{falsefalse}_{false\prod}^{i = 1} G (u_{i}, y_{i}) Q^{L} (u_{(i - 1)}, u_{i})}{\int {falsefalse}_{false\prod}^{i = 1} G (u_{i}, y_{i}) Q^{L} (u_{(i - 1)}, u_{i}) d u_{1 : k}},

and the importance distribution,

q_{1} (u_{1}) {falsefalse}_{\prod}^{i = 2} q_{i} (u_{i - 1}, u_{i}),

do not become mutually singular in the limit

d \to \infty

(or equivalently that the symmetrized Kullback–Liebler distance between these distributions does not explode with

d

).…”

Section: Some Computational Methodsmentioning

confidence: 99%

Advanced Multilevel Monte Carlo Methods

Jasra

Law

Suciu

2020

Int Statistical Rev

Self Cite

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“…More broadly speaking, the exploration of alternative proposal densities in the context of data assimilation has started only recently. See, for example, Vanden-Eijnden and Weare (2012), Morzfeld, Tu, Atkins and Chorin (2012), Van Leeuwen (2015), Llopis, Kantas, Beskos and Jasra (2018), and van Leeuwen, Künsch, Nerger, Potthast and Reich (2018). filtering Figure 3: The the initial PDF π 0 , the forecast PDF π 1 , the filtering PDF π 1 , and the smoothing PDF π 0 for a simple Gaussian transition kernel.…”

Section: Summary Of Essential Notationsmentioning

confidence: 99%

Data assimilation: The Schrödinger perspective

Reich

2019

Acta Numerica

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Data assimilation addresses the general problem of how to combine model-based predictions with partial and noisy observations of the process in an optimal manner. This survey focuses on sequential data assimilation techniques using probabilistic particle-based algorithms. In addition to surveying recent developments for discrete-and continuous-time data assimilation, both in terms of mathematical foundations and algorithmic implementations, we also provide a unifying framework from the perspective of coupling of measures, and Schrödinger's boundary value problem for stochastic processes in particular.

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“…In practice, the differential equations are represented by their discretized systems and the observations are discrete in time; therefore, we consider only the state-space model based on a discretization of the SEBM. We refer the reader to Prakasa Rao (2001), Apte et al (2007), Hairer et al (2007), Maslowski and Tudor (2013), and Llopis et al (2018) for studies about inference of SPDEs in a continuous-time setting. We discretize the SPDE Eq.…”

Section: State-space Model Representationmentioning

confidence: 99%

Joint state-parameter estimation of a nonlinear stochastic energy balance model from sparse noisy data

Lu¹,

Weitzel²,

Monahan³

2019

Nonlin. Processes Geophys.

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Abstract. While nonlinear stochastic partial differential equations arise naturally in spatiotemporal modeling, inference for such systems often faces two major challenges: sparse noisy data and ill-posedness of the inverse problem of parameter estimation. To overcome the challenges, we introduce a strongly regularized posterior by normalizing the likelihood and by imposing physical constraints through priors of the parameters and states. We investigate joint parameter-state estimation by the regularized posterior in a physically motivated nonlinear stochastic energy balance model (SEBM) for paleoclimate reconstruction. The high-dimensional posterior is sampled by a particle Gibbs sampler that combines a Markov chain Monte Carlo (MCMC) method with an optimal particle filter exploiting the structure of the SEBM. In tests using either Gaussian or uniform priors based on the physical range of parameters, the regularized posteriors overcome the ill-posedness and lead to samples within physical ranges, quantifying the uncertainty in estimation. Due to the ill-posedness and the regularization, the posterior of parameters presents a relatively large uncertainty, and consequently, the maximum of the posterior, which is the minimizer in a variational approach, can have a large variation. In contrast, the posterior of states generally concentrates near the truth, substantially filtering out observation noise and reducing uncertainty in the unconstrained SEBM.

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Particle Filtering for Stochastic Navier--Stokes Signal Observed with Linear Additive Noise

Cited by 25 publications

References 45 publications

Advanced Multilevel Monte Carlo Methods

Advanced Multilevel Monte Carlo Methods

Data assimilation: The Schrödinger perspective

Joint state-parameter estimation of a nonlinear stochastic energy balance model from sparse noisy data

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