Title: Designing low rank methods for matrices with displacement structure
In this talk, we use rational functions to design low rank methods for computing with matrices that have special displacement structures, including Toeplitz, Cauchy, Vandermonde, and more. The main workhorse of our approach is the alternating direction implicit (ADI) method. Expanding the capabilities of this classic method, we use it to explain the low rank properties of various families of matrices. We develop ADI-based low rank Sylvester and Lyapunov matrix equation solvers, low rank Poisson solvers, and superfast direct solvers for special linear systems (e.g., Toeplitz, Vandermonde). The talk covers joint work with Bernhard Beckermann, Daniel Kressner, Alex Townsend, and Daniel Rubin.