December 15, 2021 – The Mezić Group pleased to host Matt Colbrook from University of Cambridge
A recording of the talk is on our Youtube channel.
Matt Colbrook from University of Cambridge, United Kingdom is going to give a Zoom seminar on Wednesday, December 15, 2021 at 9AM PST. The zoom link is here.
Title: ResDMD: Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems
Abstract: Koopman operators are infinite-dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continuous spectra and infinite-dimensional invariant subspaces, making computing their spectral information a considerable challenge. This paper describes data-driven algorithms with rigorous convergence guarantees for computing spectral information of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), which provides the first scheme for computing the spectra and pseudospectra of general Koopman operators from snapshot data without spectral pollution. Using the resolvent operator and ResDMD, we compute smoothed approximations of spectral measures associated with measure-preserving dynamical systems. We prove explicit convergence theorems for our algorithms, which can achieve high-order convergence even for chaotic systems, when computing the density of the continuous spectrum and discrete spectrum. We demonstrate our algorithms on the tent map, Gauss iterated map, nonlinear pendulum, double pendulum, Lorenz system, and an 11-dimensional extended Lorenz system. Finally, we provide kernelized variants of our algorithms for dynamical systems with a high-dimensional state-space. This allows us to compute the spectral measure associated with the dynamics of a protein molecule that has a 20,046-dimensional state-space and compute nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number greater than 10^5 that has a 295,122-dimensional state-space. This talk is based on joint work with Alex Townsend and our preprint can be found here: https://arxiv.org/abs/2111.14889.
Short Bio: Matthew Colbrook is a Junior Research Fellow at Trinity College Cambridge and a Fondation Sciences Mathématiques de Paris Postdoctoral Fellow at École Normale Supérieure. He holds a PhD from the Department of Applied Mathematics and Theoretical Physics, University of Cambridge (2020). His research is centred on numerical analysis and foundations of computation in infinite-dimensional spectral problems, PDEs, and deep learning/neural networks for scientific computation, as well as a framework for determining the boundaries of what is and what is not computationally possible. He is a recipient of the IMA Lighthill-Thwaites Prize, the Smith-Knight prize and the Mayhew prize.