Abstract: We propose the adaptive spectral Koopman (ASK) method to solve nonlinear autonomous dynamical systems. This novel numerical method leverages the spectral-collocation method and properties of the Koopman operator to obtain the solution of a dynamical system. Specifically, this solution is represented by the Koopman operator’s eigenfunctions, eigenvalues, and Koopman modes. Unlike conventional time evolution algorithms such as Euler’s scheme and the Runge-Kutta scheme, ASK is mesh-free and more flexible when evaluating the solution. Numerical experiments demonstrate high accuracy of ASK for solving both ordinary and partial differential equations. Further, ASK enables new designs of uncertainty quantification (UQ) methods, which can be much faster than state-of-the-art UQ methods. Finally, we will illustrate ASK's capability of solving optimization problems based on the gradient flow formula.
Speaker’s Bio: Xiu Yang obtained his bachelor's and master's degree from Peking University, and his Ph.D. from Brown University. He joined Lehigh University from the Pacific Northwest National Laboratory (PNNL) where he was a scientist since 2016. His research has been centered around modern scientific computing including uncertainty quantification, multi-scale modeling, physics-informed machine learning, and data-driven scientific discovery. Xiu has been applying his methods to various research areas such as fluid dynamics, hydrology, biochemistry, soft material, climate modeling, energy storage, and power grid system. Currently, he is focusing on UQ in quantum computing algorithms and machine learning methods for scientific computing. He received a Faculty Early Career Development Program Award from the National Science Foundation in 2022 and Outstanding Performance Award from PNNL in 2015 and 2016. Xiu also served on the Department of Energy’s applied mathematics visioning committee in 2019.
Last Updated: May 1, 2023 - 10:16 am