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AEM Seminar: Koopman Operator Theory in Fluid Mechanics

Igor Mezic, Professor, Department of Mechanical Engineering, University of California, Santa Barbara

2:30 PM on 2016-04-22

319 Akerman Hall


There is long history of contributions to modal representation of nonlinear fluid flows, including global modes, triple decomposition and Proper Orthogonal Decomposition. Recently, a spectral decomposition relying on Koopman operator theory has attracted interest in the fluid mechanics community. Part of the attractiveness of the Koopman operator approach stems from ability to compute the modes from relatively simple algorithms such as the Dynamic Mode Decomposition (DMD). To make a connection between the two, we show an explicit relationship between a basic version of the Dynamic Mode Decomposition (DMD) and the Koopman Mode Decomposition (KMD) of dynamical systems, that allows for estimates of validity of approximation of Koopman modes by DMD modes, and in the process introduce the notion of Generalized Laplace analysis that enables the underlying calculations. We also discuss the notion of exact solutions and Resolvent Mode Decomposition in connection with Koopman Modes. In addition, traditionally the accuracy of decomposition methods has been tested by comparing the flow with its modal approximation in quadratic norm sense. We will discuss an alternative approach that incorporates recent advances in characterizing kinematics of aperiodic-in-time flows on finite time-scales by using the concept of mesohyperbolicity. This coupling of Koopman Mode Decomposition and Mesohyperbolicity Theory enables us to start utilizing dynamical systems theories of mixing in the context of flows with non-periodic time dependence.

Bio:

Professor Mezic works on operator-theoretic methods in nonlinear dynamical systems and control theory and their applications in fluid dynamics, energy efficient design and operations and complex systems dynamics. He did his Dipl. Ing. in Mechanical Engineering in 1990 at the University of Rijeka, Croatia and his Ph. D. in Applied Mechanics at the California Institute of Technology. Dr. Mezic was a postdoctoral researcher at the Mathematics Institute, University of Warwick, UK in 1994-95. From 1995 to 1999 he was a member of Mechanical Engineering Department at the University of California, Santa Barbara where he is currently a Professor. In 2000-2001 he has worked as an Associate Professor at Harvard University in the Division of Engineering and Applied Sciences. He won the Alfred P. Sloan Fellowship, NSF CAREER Award from NSF and the George S. Axelby Outstanding Paper Award on "Control of Mixing" from IEEE. He also won the United Technologies Senior Vice President for Science and Technology Special Achievement Prize in 2007. He was an Editor of Physica D: Nonlinear Phenomena and an Associate Editor of the Journal of Applied Mechanics and SIAM Journal on Control and Optimization. Dr. Mezic is the Director of the Center for Energy Efficient Design and Head of Buildings and Design Solutions Group at the Institute for Energy Efficiency at the University of California, Santa Barbara. For more information on Prof. Mezic’s research group, please visit www.engr.ucsb.edu/~mgroup.


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