Prof. Igor Mezic (UCSB Mechanical Engineering)

Title: Uncertainty Analysis: A Dynamical Systems Approach

Abstract: Uncertainty is measured in many different ways in different fields. The most popular ways of assessing uncertainty are through variance of a probability distribution or its information-theoretic entropy. There is however, a different and very natural notion of uncertainty of a probability distribution: uncertainty can be thought of as a "distance to a certain distribution". We will take this notion and formalize it to show how it leads to measures of uncertainty that have interesting, computable properties. We will then describe how uncertainty evolves in time under the action of a dynamical system, where no assumptions of linearity or Gaussianity are made, using an operator-theoretic approach. This formalism is applied to study dynamical systems that have a specific, horizontal-vertical graph structure and prove that in such systems propagation of uncertainty is tightly structured, perhaps explaining their prevalence in engineered and biological systems. Finally, we discuss approaches to computation of uncertainty and present a modulation of Quasi-Monte Carlo method using a construction that samples the space of parameters or initial conditions using uniformly ergodic dynamical systems that preserve the specified prior. The combination of techniques described above provides us with ability to analyze propagation of uncertainty in dynamical systems with large dimensionality. We will discuss a variety of applications, including gene regulation networks.