Room: 5-314
Speaker Name:
Hessam Babaee
Affiliation:
Department of Mechanical Engineering & Material Sciences, University of Pittsburgh
Abstract:
High-dimensional tensor differential equations (TDEs) are omnipresent in science and engineering. Example applications include the Schrödinger equation, probability density function transport equations in turbulent combustion, the Fokker-Planck equation, the Boltzmann transport equation, and the Hamilton-Jacobi-Bellman equation, among others. However, performing computational tasks involving high-dimensional tensors, or even storing them, suffers from the curse of dimensionality: the total number of elements of a tensor increases exponentially as the dimension grows. Various tensor low-rank approximations have been developed to mitigate this issue by leveraging multi-dimensional correlations. These dimen- sion reduction techniques aim to decrease the total number of tensor elements while allowing for a controllable loss of accuracy. In this seminar, we review the recent advancements in dynamical low-rank approximation (DLRA), which provides a rigorous mathematical frame- work for solving matrix differential equations (MDEs) and TDEs on low-rank manifolds. We particularly focus on the computational cost issues of solving DLRA equations. We present a new formulation for solving nonlinear MDEs and TDEs based on oblique projection onto low-rank manifolds. Several applications are presented, including turbulent combustion, Fokker-Planck equations, sensitivity analysis, and uncertainty quantification.
Biography:
Hessam Babaee is currently an Associate Professor in the Department of Mechanical Engineering and Materials Science at the University of Pittsburgh. He earned his Ph.D. in Mechanical Engineering and a Master’s in Applied Mathematics, both from Louisiana State University, in 2013. He then joined the Mechanical Engineering Department at MIT for his postdoctoral research before moving to the University of Pittsburgh in January 2017. His research focuses on developing low-rank approximation techniques for high-dimensional dynamical systems. His work has been funded by many organizations, including the National Aeronautics and Space Administration (NASA), the National Science Foundation (NSF), the National Institutes of Health (NIH), and the Air Force Office of Scientific Research (AFOSR).