Applied Mathematics Colloquium: Dr Michal Branicki

Title: Lagrangian uncertainty quantification and information inequalities in stochastic flows 

Abstract: We develop a systematic information-theoretic framework for quantification and mitigation of error in probabilistic path-based (Lagrangian) predictions which are obtained from dynamical systems generated by uncertain (Eulerian) vector fields. This work is motivated by the desire to improve Lagrangian predictions in complex dynamical systems based either on analytically simplified or data-driven models. We derive a hierarchy of general information bounds on the uncertainty in estimates of statistical observables $E^\nu[f]$, evaluated on trajectories of the approximating dynamical system, relative to the ‘true’ observables $E^\mu[f]$ in terms of certain $\varphi$-divergencies  $D(\mu||\nu)$ which quantify discrepancies between probability measures $\mu$ associated with the original dynamics and their approximations $\nu$. We then derive bounds on $D(\mu||\nu)$  itself in terms of the Eulerian fields. This framework provides a rigorous way for quantifying and mitigating uncertainty in Lagrangian predictions due to Eulerian model error. Links to uncertainty quantification in Data Assimilation techniques will also be mentioned.