A differential geometric approach to nonlinear filtering: the projection filter

Scientific computations for the quantification, estimation and prediction of uncertainties for ocean dynamics are developed and exemplified. Primary characteristics of ocean data, models and uncertainties are reviewed and quantitative data assimilation concepts defined. Challenges involved in realistic data-driven simulations of uncertainties for four-dimensional interdisciplinary ocean processes are emphasized. Equations governing uncertainties in the Bayesian probabilistic sense are summarized. Stochastic forcing formulations are introduced and a new stochastic-deterministic ocean model is presented. The computational methodology and numerical system, Error Subspace Statistical Estimation, that is used for the efficient estimation and prediction of oceanic uncertainties based on these equations is then outlined. Capabilities of the ESSE system are illustrated in three data-assimilative applications: estimation of uncertainties for physical-biogeochemical fields, transfers of ocean physics uncertainties to acoustics, and real-time stochastic ensemble predictions with assimilation of a wide range of data types. Relationships with other modern uncertainty quantification schemes and promising research directions are discussed.

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“…Other methodologies are based on moment approximations or finitedimensional projections of densities onto manifolds (e.g. [5,6]). …”

Section: Relationships To Other Approaches and Promising Directionsmentioning

confidence: 99%

“…Using Eqs. (5) and (6) of Section 3.1.1, spatially discrete stochastic forcings are added, only in the equations which are prognostic.…”

Section: A2 Stochastic Modelsmentioning

confidence: 99%

Uncertainty estimation and prediction for interdisciplinary ocean dynamics

Lermusiaux

2006

Journal of Computational Physics

160

121

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“…Many different methods were proposed, cf. [7,11,14,18,19,32,37,42]. In the present paper we introduce a solution method for the Zakai equation based on the Monte-Carlo simulation of a robust, recursive Feynman-Kac formula.…”

Section: Introductionmentioning

confidence: 99%

Robust Parameter Estimation for Stochastic Differential Equations

Deck

Theting

2004

Acta Appl Math

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We consider the estimation of parameters in stochastic differential equations (SDEs). The problem is treated in the setting of nonlinear filtering theory with a degenerate diffusion matrix. A robust stochastic Feynman-Kac representation for solutions of SDEs of Zakai-type is derived. It is verified that these solutions are conditional densities for the conditional measures defined by degenerate filtering problems. We show that the corresponding estimator for the parameters is robust in the following sense: It depends continuously on both the measurement path and on the intensity of the measurement noise. An algorithm based on a Monte-Carlo approach is given for the practical application of the estimator, and numerical results are reported. Mathematics Subject Classifications (2000): Primary: 62M05, 62M20; secondary: 62F15.

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“…Note, that a differential geometric filtering approach for continuous-time problems which also aims at the minimization of a distance measure between the exact and approximate density functions has been published in [5], [6]. In contrast to that work, the approach presented in this paper introduces a progression approach which defines continuous moment variations and thus continuous variations of the density parameters for discrete-time systems.…”

Section: Motivation Of the Moment-based Prediction Step For Expomentioning

confidence: 97%

“…These parameterizable densities are often chosen as exponential densities with polynomial exponents [5], [6]. The case of Gaussian density approximations corresponds to finding optimal parameters of an exponential density with second order polynomial exponent.…”

mentioning

confidence: 99%

Moment-Based Prediction Step for Nonlinear Discrete-Time Dynamic Systems Using Exponential Densities

Rauh

Hanebeck²

Proceedings of the 44th IEEE Conference on Decision and Control

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In this paper, an efficient approach for a momentbased Bayesian prediction step for both linear and nonlinear discrete-time dynamic systems using exponential densities with polynomial exponents is proposed. The exact solution of the prediction step is approximated by an exponential density which minimizes the Kullback-Leibler distance. Compared to other approaches, the user of this procedure can specify the approximation quality by controlling the deviation between the moments of the exact and the approximated solution. Furthermore, this algorithm can also be used for the adaptation of the order of the exponential densities either to improve the approximation quality or to reduce the computational effort.

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A differential geometric approach to nonlinear filtering: the projection filter

Cited by 41 publications

References 17 publications

Uncertainty estimation and prediction for interdisciplinary ocean dynamics

Uncertainty estimation and prediction for interdisciplinary ocean dynamics

Robust Parameter Estimation for Stochastic Differential Equations

Moment-Based Prediction Step for Nonlinear Discrete-Time Dynamic Systems Using Exponential Densities

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