Branicki M., Chen N., Majda A.J.,   Non-Gaussian Test Models for Prediction and State Estimation with Model Errors, Chinese Ann. Math. (special issue in honor of Jacques-Louis Lions) 34B(1), 29–64 (2013)
Branicki M., Majda, A.J.,  Quantifying uncertainty for statistical predictions with model errors in non-Gaussian systems with intermittency,  Nonlinearity 25, 2543-2578 (2012)
Branicki M., Majda, A.J.,  Fundamental Limitations of Polynomial Chaos for Uncertainty Quantification in Systems with Intermittent Instabilities,  Comm. Math. Sci. 11(1), (2012)
Majda, A.J., Branicki M., Lessons in Uncertainty Quantification for Turbulent Dynamical Systems, Discrete Contin. Dynam. Systems  32(9), 3133-3231, (2012)
Majda, A.J., Branicki, M., Frenkel, Y.,  Improving Complex Models Through Stochastic Parameterization and Information Theory,  ECMWF Proceedings, Representing Model Uncertainty and Error in Numerical Weather and Climate Prediction Models, 121-136(2011)
Majda A.J., Branicki M., Turbulent Dynamical Systems in Climate Science
Branicki M., Majda, A.J., Quantifying Filter Performance for Turbulent Dynamical  
                                          Systems through Information Theory

Turbulent dynamical systems involve dynamics in a high-dimensional phase space with a large number of positive Lyapunov exponents and intermittent energy transfers across a wide range of spatio-temporal scales. Discretizations of such systems are ubiquitous in applications where, despite the incomplete knowledge of the underlying dynamics, statistical ensemble prediction and real-time state estimation from coarse-grained observations are needed. Many nonlinear, multi-scale systems  display a subtle  interplay between sensitivity to external  perturbations and the nature of chosen approximations for the unresolved processes. The inevitable presence of intrinsic model errors and the `curse of small ensemble size’ makes the assessment of predictive skill  for the coarse-grained, long-time trends in turbulent systems a serious challenge.

        In a series of papers with A. Majda we developed a statistical-stochastic framework to address these issues in complex partially observed systems with model error. This novel framework blends techniques from information theory, stochastic dynamical systems, and fluctuation-dissipation theory for forced dissipative systems. Information theory provides natural measures for  quantification of both the initial-value predictability and sensitivity to forced perturbations of imperfect reduced-order dynamical models.