Quantifying Bayesian filter performance for turbulent dynamical systems through information theory

M. Branicki, A. J. Majda

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)901-978
Number of pages78
JournalCommunications in Mathematical Sciences
Volume12
Issue number5
DOIs
StatePublished - 2014

Fingerprint

Information theory
Information Theory
Dynamical systems
Dynamical system
Filter
Model Error
Imperfect
Information Measure
Forecast
High-dimensional
Filtering
Error statistics
Gaussian Filter
Advection-diffusion Equation
Data Assimilation
Information Criterion
Kalman Filtering
Observability
Advection
Mean square error

Keywords

  • Information theory
  • Kalman filter
  • Turbulent dynamical systems
  • Uncertainty quantification

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Quantifying Bayesian filter performance for turbulent dynamical systems through information theory. / Branicki, M.; Majda, A. J.

In: Communications in Mathematical Sciences, Vol. 12, No. 5, 2014, p. 901-978.

Research output: Contribution to journalArticle

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