Igor Mezić is the 2021 recipient of the J.D. Crawford Prize. The award will be presented virtually at the SIAM Conference on Application of Dynamical Systems (DS21), to be held in a virtual format May 23 – 27, 2021. The prize is awarded to Mezić for his paper, “Spectrum of the Koopman Operator, Spectral Expansions in Functional Spaces, and State-Space Geometry”.
The SIAM Activity Group on Dynamical Systems (SIAG/DS) awards the J. D. Crawford Prize every two years to one individual for recent outstanding work on a topic in nonlinear science, as evidenced by a publication in English in a peer-reviewed journal within the four calendar years preceding the award year. The term “nonlinear science" is used in the spirit of the SIAG on Dynamical Systems conferences. Specifically, it includes dynamical systems theory and its applications as well as experiments, computations, and simulations.
Dr. Igor Mezić is a Full Professor at the Department of Mechanical Engineering, and the Department of Mathematics at the University of California, Santa Barbara (UCSB). He is also the Director of the Center for Energy Efficient Design and Head of Buildings and Design Solutions Group at the Institute for Energy Efficiency at UCSB. Dr. Mezic works on operator-theoretic methods in nonlinear dynamical systems and control theory and their applications in fluid dynamics, energy efficient design, network security 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 worked as an Associate Professor at Harvard University in the Division of Engineering and Applied Sciences.
He has won numerous prizes for his research, among them the Alfred P. Sloan Fellowship in Mathematics, the National Science Foundation CAREER Award, 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 for his contribution to a variety of engineering technologies. He gave numerous plenary lectures at international conferences and was an Editor of Physica D: Nonlinear Phenomena and a number of other research publications. He is a SIAM Fellow and a Fellow of the American Physical Society.
Q: Why are you excited to receive the award of the J. D. Crawford Prize?
A: The fact that the dynamical systems community deems my work worthy of the Crawford prize is wonderful. The depth and breadth of the dynamical systems research is immense. Much of it is of top technical quality, while being of wide impact in sciences and engineering. Thus the value of the prize is heightened just by the sheer quality and volume of the competition for it. On a personal note, I participate in every SIAM DS meeting - except if illness prevents me from that - and thoroughly enjoy the community. That makes receiving the award even more pleasant.
Q: Could you tell us a bit about the research that won you the prize?
A: At the beginning of my research career, at Caltech in early 1990's, I was at the same time fascinated by the geometrical approach to dynamical systems, and frustrated by its limitations in yielding results in higher dimensions and non-perturbative problems. I started thinking about the way to resolve these limitations while retaining the power of the geometrical approach and its fundamental concepts - e.g. invariant sets, stable and unstable manifolds. I found that the classic Koopman operator theoretic approach to measure-preserving dynamical systems offers some answers. But the method - at that stage - was not developed for the all-present dissipative systems, could not be used to identify important geometrical objects such as stable and unstable manifolds, and did not have a solid computational basis.
Additionally, even after all the work we have been doing since, before my latest work, we did not have a clear classification of which types of computed Koopman operator spectra obtained from simulations or experiments for dissipative systems correspond to which type of a dynamical system (e.g. spectra for a system globally stable to a fixed point vs. globally stable to a toroidal attractor). The work I was awarded for, "Mezić I. Spectrum of the Koopman operator, spectral expansions in functional spaces, and state-space geometry. Journal of Nonlinear Science. 2019 Dec 6:1-55.", resolved many of those issues. For example, the standard Hilbert space setting for operator-theoretic analysis and computation was sufficient in the classical, measure-preserving Koopman setting. But it is inadequate in the setting of dissipative systems. This influences practical issues such as computations, because in these we typically need a basis for a space we are working on, and that basis differs depending on the selected space. In the same paper, stable, unstable, and center manifolds were reformulated as operator-theoretic objects, via level sets of eigenfunctions. This enables their computation using computational linear algebra.
Another aspect that the work resolved is the nature of the spectrum of dissipative dynamical systems in the newly constructed functional spaces. Once the type of the spectrum is known, one can recognize the system from measuring its spectrum. This is akin to the fact that a harmonic process can be recognized by the fact that it is represented by a single peak at a single frequency in the output of the oscilloscope using the classical Fourier spectrum. In the Koopman operator setting I showed that analogous results hold by looking at the distribution of eigenvalues of the Koopman operator in the complex plane. The connection with data-driven analysis and modern artificial intelligence methods for learning of dynamical systems is immediate. The contribution builds on a large body of work with my students, postdocs and collaborators pursued since my thesis and I'd like to thank them all for their insights and collaboration.
Q: What does your work mean to the public?
A: The methodology that was developed is a fertile ground for applications in data analysis, and AI that are already being pursued in a variety of fields. Fluid mechanics, network security, power grid operations, building energy efficiency, soft robotics, and neuroscience are just some that come to mind. In all of these, the algorithms and artificial intelligence implementations based on Koopman operator theory already yielded useful practical results. For example, we have been able to reduce energy use at the UCSB campus building by 20 percent using these methods. We have also developed artificial intelligence methodologies for network security that are currently deployed with large commercial and government entities, protecting their networks and data. The specific work the award is for addresses fundamental issues in computational and data analysis aspects of such applications that will enable more efficient computation and implementation.
Q: What does being a member of SIAM mean to you?
A: I have wide research interests, but the Society for Industrial and Applied Mathematics is the key professional organization in the field that my research is most closely associated with. This means that in the SIAM community I have friends and colleagues that I can work with, learn from, and exchange opinions with without an effort of breaking down the communication barrier. It means I have access to a host of research journals edited by colleagues that make them high quality, and timely in the delivery of the research material without the traps of commercial publishing. It also provides for research camaraderie at conferences, where learning and delivering knowledge on new aspects of our fields is exciting and stimulating. Overall, SIAM played a huge role in my career so far (being elected a Fellow a few years ago was a true highlight) and will undoubtedly stay so for the rest of it.
--Article by SIAM News