Colloquium: Hongkai Zhao

PLEASE NOTE: 4:00--REGULAR TIME

Speaker: Hongkai Zhao (University of California at Irvine)

Title: Approximate separability of Green’s function and intrinsic complexity of differential operators

Abstract: Approximate separable representation of the Green’s functions for differential operators is a fundamental question in the analysis of differential equations and development of efficient numerical algorithms. It can reveal intrinsic complexity, e.g., Kolmogorov n-width or degrees of freedom of the corresponding differential equation. Computationally, being able to approximate a Green’s function as a sum with few separable terms is equivalent to the existence of low rank approximation of the discretized system which can be explored for matrix compression and fast solution techniques such as in fast multiple method and direct matrix inverse solver. In this talk, we will mainly focus on Helmholtz equation in the high frequency limit for which we developed a new approach to study the approximate separability of Green’s function based on an geometric characterization of the relation between two Green's functions and a tight dimension estimate for the best linear subspace approximating a set of almost orthogonal vectors. We derive both lower bounds and upper bounds and show their sharpness and implications for computation setups that are commonly used in practice. We will also make comparisons with other types of differential operators such as coercive elliptic differential operator with rough coefficients in divergence form and hyperbolic differential operator. This is a joint work with Bjorn Engquist.

Faculty host: Yanzhao Cao

Brief Description of the Speaker’s Academic and Professional Achievements/Credentials:

Prof. Hongkai Zhao works in computational and applied mathematics that includes modeling, analysis and developing numerical methods for problems arising from science and engineering. He has published over 90 papers in peer review journals, and has received several prestigious awards such as Alfred P. Sloan Fellowship, Feng Kang prize in Scientific Computing, etc. He serves on the editorial board of SIAM Multiscale Modeling and Simulation, Progress in Mathematics Annals of Mathematical Sciences and Applications Journal of Computational Mathematics Computational & Applied Mathematics, etc. During 2010-2013, he serves as the chair of the math department at UC Irvine.