Non-Gaussian Test Models for Prediction and State Estimation with Model Errors

Branicki, Michał; Chen, Nan; Majda, Andrew J.

doi:10.1007/978-3-642-41401-5_4

Cited by 5 publications

(5 citation statements)

References 36 publications

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“…Intermittency is an important physical phenomena. Exactly solvable test models as a test bed for the prediction and UQ strategy [54,48,57] including information barriers are discussed extensively in models ranging from linear stochastic models to nonlinear models with intermittency in the research expository article [56] as well as in [7,8]. Some more sophisticated applications are mentioned next in Section 1.4.…”

Section: Modeling Stages Math and Computational Toolsmentioning

confidence: 99%

Strategies for Reduced-Order Models for Predicting the Statistical Responses and Uncertainty Quantification in Complex Turbulent Dynamical Systems

Majda¹,

Qi²

2018

SIAM Rev.

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Turbulent dynamical systems characterized by both a high-dimensional phase space and a large number of instabilities are ubiquitous among many complex systems in science and engineering including climate, material, and neural science. The existence of a strange attractor in the turbulent systems containing a large number of positive Lyapunov exponents results in a rapid growth of small uncertainties from imperfect modeling equations or perturbations in initial values, requiring naturally a probabilistic characterization for the evolution of the turbulent system. Uncertainty quantification (UQ) in turbulent dynamical systems is a grand challenge where the goal is to obtain statistical estimates such as the change in mean and variance for key physical quantities in their nonlinear responses to changes in external forcing parameters or uncertain initial data. In the development of a proper UQ scheme for systems of high or infinite dimensionality with instabilities, significant model errors compared with the true natural signal are always unavoidable due to both the imperfect understanding of the underlying physical processes and the limited computational resources available through direct Monte-Carlo integration. One central issue in contemporary research is the development of a systematic methodology that can recover the crucial features of the natural system in statistical equilibrium (model fidelity) and improve the imperfect model prediction skill in response to various external perturbations (model sensitivity).A general mathematical framework to construct statistically accurate reduced-order models that have skill in capturing the statistical variability in the principal directions with largest energy of a general class of damped and forced complex turbulent dynamical systems is discussed here. There are generally three stages in the modeling strategy, imperfect model selection; calibration of the imperfect model in a training phase; and prediction of the responses with UQ to a wide class of forcing and perturbation scenarios. The methods are developed under a universal class of turbulent dynamical systems with quadratic nonlinearity that is representative in many applications in applied mathematics and engineering. Several mathematical ideas will be introduced to improve the prediction skill of the imperfect reduced-order models. Most importantly, empirical information theory and statistical linear response theory are applied in the training phase for calibrating model errors to achieve optimal imperfect model parameters; and total statistical energy dynamics are introduced to improve the model sensitivity in the prediction phase especially when strong external perturbations are exerted. The validity of general framework of reduced-order models is demonstrated on instructive stochastic triad models. Recent applications to two-layer baroclinic turbulence in the atmosphere and ocean with combinations of turbulent jets and vortices are also surveyed. The uncertainty quantification and statistical response for th...

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Section: Modeling Stages Math and Computational Toolsmentioning

confidence: 99%

Strategies for Reduced-Order Models for Predicting the Statistical Responses and Uncertainty Quantification in Complex Turbulent Dynamical Systems

Majda¹,

Qi²

2018

SIAM Rev.

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“…The natural way to quantify the error in the recovered PDF related to the truth is through an information measure, namely the relative entropy (or Kullback-Leibler divergence) [63,52,64,65,66]. The relative entropy is defined as L is increased to 100, the error in the recovered PDFs becomes insignificant.…”

Section: Performance Tests With Highly Non-gaussian Featuresmentioning

confidence: 99%

Efficient statistically accurate algorithms for the Fokker–Planck equation in large dimensions

Chen

Majda

2018

Journal of Computational Physics

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Solving the Fokker-Planck equation for high-dimensional complex turbulent dynamical systems is an important and practical issue. However, most traditional methods suffer from the curse of dimensionality and have difficulties in capturing the fat tailed highly intermittent probability density functions (PDFs) of complex systems in turbulence, neuroscience and excitable media. In this article, efficient statistically accurate algorithms are developed for solving both the transient and the equilibrium solutions of Fokker-Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures. The algorithms involve a hybrid strategy that requires only a small number of ensembles. Here, a conditional Gaussian mixture in a high-dimensional subspace via an extremely efficient parametric method is combined with a judicious non-parametric Gaussian kernel density estimation in the remaining low-dimensional subspace. Particularly, the parametric method provides closed analytical formulae for determining the conditional Gaussian distributions in the high-dimensional subspace and is therefore computation-ally efficient and accurate. The full non-Gaussian PDF of the system is then given by a Gaussian mixture. Different from the traditional particle methods, each conditional Gaussian distribution here covers a significant portion of the high-dimensional PDF. Therefore a small number of ensembles is sufficient to recover the full PDF, which overcomes the curse of dimensionality. Notably, the mixture distribution has a significant skill in capturing the transient behavior with fat tails of the high-dimensional non-Gaussian PDFs, and this facilitates the algorithms in accurately describing the intermittency and extreme events in complex turbulent systems. It is shown in a stringent set of test problems that the method only requires an order of O(100) ensembles to successfully recover the highly non-Gaussian transient PDFs in up to 6 dimensions with only small errors.

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“…The present information-theoretic framework allows for a systematic optimization of the information content in the filter estimates, as well as identification of information barriers [50,58,13,10] in the imperfect Kalman filters. As discussed in Section 2.2.1, the quantification of different facets of filter skill in the present superensemble framework relies on the entropy of the filter error S(u u u m −ū u u m|m ), the mutual information M (u u u m ,ū u u m|m ), and the relative entropy P(π(u u u m ),π f (ū u u m|m )), which provide measures of an 'information distance' between the truth u u u m and its filter estimateū u u m|m .…”

Section: Information Optimization Of Imperfect Kalman Filtersmentioning

confidence: 99%

Quantifying Bayesian filter performance for turbulent dynamical systems through information theory

Branicki

Majda

2014

Communications in Mathematical Sciences

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Abstract. Incomplete knowledge of the true dynamics and its partial observability pose a notoriously difficult problem in many scientific applications which require predictions of high-dimensional dynamical systems with instabilities and energy fluxes across a wide range of scales. In such cases assimilation of real data into the modeled dynamics is necessary for mitigating model error and for improving the stability and predictive skill of imperfect models. However, the practically implementable data assimilation/filtering strategies are also imperfect and not optimal due to the formidably complex nature of the underlying dynamics. Here, the connections between information theory and the filtering problem are exploited in order to establish bounds on the filter error statistics, and to systematically study the statistical accuracy of various Kalman filters with model error for estimating the dynamics of spatially extended, partially observed turbulent systems. The effects of model error on filter stability and accuracy in this high-dimensional setting are analyzed through appropriate information measures which naturally extend the common path-wise estimates of filter performance, like the mean-square error or pattern correlation, to the statistical superensemble setting that involves all possible initial conditions and all realizations of noisy observations of the truth signal. Particular emphasis is on the notion of practically achievable filter skill which requires trade-offs between different facets of filter performance; a new information criterion is introduced in this context. This information-theoretic framework for assessment of filter performance has natural generalizations to Kalman filtering with non-Gaussian statistically exactly solvable forecast models. Here, this approach is utilized to study the performance of imperfect, reduced-order filters involving Gaussian forecast models which use various spatio-temporal discretizations to approximate the dynamics of the stochastically forced advection-diffusion equation; important examples in this configuration include effects of biases due to model error in the filter estimates for the mean dynamics which are quantified through appropriate information measures.

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Non-Gaussian Test Models for Prediction and State Estimation with Model Errors

Cited by 5 publications

References 36 publications

Strategies for Reduced-Order Models for Predicting the Statistical Responses and Uncertainty Quantification in Complex Turbulent Dynamical Systems

Strategies for Reduced-Order Models for Predicting the Statistical Responses and Uncertainty Quantification in Complex Turbulent Dynamical Systems

Efficient statistically accurate algorithms for the Fokker–Planck equation in large dimensions

Quantifying Bayesian filter performance for turbulent dynamical systems through information theory

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