A strongly nonlinear system is a chaotic dynamical system which exhibits energy transfer between various scales in time and space. The well-known examples of this behavior commonly appear in engineering and geophysical turbulence. In this project, we work on capturing relatively low-dimensional but statistically accurate models of these systems with the aid of data. The end goal is to use these models, instead of very costly physics-based models, to characterize and predict extreme events (e.g. heat waves), study the statistical response to external output (e.g. climate trends), and synthesize long-time control strategies (e.g. mean drag reduction).
Why do we hear so much about data? ... data has become instrumental in dealing with systems that have large dimensions and contain uncertainy. The traditional approach in dynamical systems theory has been centered around geometric analysis in the state space, but a lot of systems that we are interested in right now have either too many degrees of freedom to model directly (e.g. climate, turbulent flows and chemical kinetics) or lack reliable models for state-space analysis (e.g. biological and social networks). Data helps us detect and exploit low-dimensional structures in the former group, or find surrogate but amenable models for systems of the latter. Two sets of techniques have contributed the most to data-driven analysis and control in the recent years: 1-operator-theoretic methods and 2- machine learning. So far our work has mostly benifted from the Koopman-operator persepctive but there is much promise in ideas from machine learning and their connection to operator-theoeretic viewpoint.
The goal of this project is to devise and analyze numerical algorithms that enable data-driven analysis and control of dynamical systems. For example, Dynamic Mode Decomposition (DMD) is a linear data decomposition algorithm that has become the primary tool for computing the Koopman eigenfunctions (i.e. linearly evolving coordinates) in simple nonlinear systems. In previous works, we have established rigorous results on the connection between this algorithm, and some other signal processing methods, with operator-theoretic viewpoint for nonlinear systems. We are interested in improving the performance of numerical toolbox for more efficient and robust data-driven modeling, for example by including optimal sampling of state and inputs.
Mixing is a crucial part of many processes related to fluids like movement of oil slicks in the ocean, processing of polymers and metals, and biodmedical testing. Despite its prevalence, the process of mixing in flows, even in the absence of diffusion and particle-size effects, is not fully understood. Most of rigorous analysis in the field of advective mixing comes from the theory of chaotic advection which treats the fluid flow as a dynamical system with trajectories that describe the paths of tracers. A strong theme of our work is to understand mixing by combining the fluid mechanical apsect (i.e. Navier-Stokes dynamics) with the methods of chaotic advection. For example, this research has led to extension of classical Prandtl-Batchelor theory to unsteady 2D flows which predicts that at high Reynolds the rotational structures in the flow resemble rigid body rotation. Interstingly, resonance arguments from chaotic advection show that this constant-vorticity condition implies that mixing in the core of those flows would be weaker than wall-adjacent areas, which is radically different from mixing in low-Reynolds flows.
On a more applied side, we collaborated with scientists in Marine Science Institute to devise a framework for prediction of oil spill movement on the ocean. This problem was motivated by the 2015 Refugio oil spill in northern Santa Barbara county.