Data Driven Uncertainty Quantification for Complex Turbulent Systems

There are numerous contexts where one wishes to describe the hidden state of a randomly evolving system, based upon accessible time intermittent noisy observations. Filtering (or Bayesian data assimilation) refers to the scientifically sound methodology that carries this out sequentially in time through combining dynamical system model along with observational data in order to estimate the uncertainty associated to the system state; this is a form of data driven uncertainty quantification. Due to the practical relevance and theoretical intricacy, there have been tremendous efforts to solve the filtering problem. Yet, in contrast to the low dimensional case in which some considerable progress has been made, it turns out particularly difficult to achieve accurate filtering for complex dynamical systems in high dimension, such as those arising in turbulence. Because many important dynamical system models are inherently described by large number of degrees of freedom, how to overcome the high dimensional obstacles is a fundamental question in this area.In its full form, filtering requires to describe a time-evolving probability distribution on the system state, conditioned on data, and effective low-dimensional representation of the desired distribution is challenging for turbulent system. Nonetheless recent advances by Andrew Majda and coworkers have demonstrated the possibility of filtering based on carefully chosen simplifications of the underlying system, even when they are modelled simple enough to enable closed form filters to be developed. This research direction of the reduced model approach for filtering is clearly in a very early stage and many associated questions remain unanswered - to name a few, for instance, why this heuristic argument works, how to tune the adaptive parameters, how to develop other simplified models, how the misspecification between model and data affects the filtering solution in a long time. Therefore, this body of work gives rise to a timely opportunity of making valuable contributions to the emergence of a new paradigm of filtering.The proposal aims to answer these questions through analysis, through derivation of new methods in the similar spirit, and through careful numerical experiments. In particular, the goal of the project is meticulous modeling of approximate systems in a moderate dimension utilizing a rigorous theoretical analysis of the original system so that both dynamical systems share key statistical features. The feasibility of this research is supported by our studies on nontrivial system behavior of one-dimensional turbulence prototype, and by our recent works on derivation and analysis of simplified filters.