The computation of extremal eigenvalues of large, sparse matrices has proven to be one of the most important problems in numerical linear algebra. Krylov subspace methods are a powerful class of techniques for this problem, most notably the Arnoldi process for non-symmetric matrices and the Lanczos method for symmetric matrices. The theory of convergence for the Lanczos method is well understood, but much less is known about uniform error estimates for a specific number of iterations. Uniform estimates are increasingly relevant, as, for large matrices, the number of iterations is often orders of magnitude less than the dimension of the matrix. In this talk, we will introduce the Lanczos method, present existing error estimates, and give a detailed discussion of recent uniform error estimates.
Short Bio Provided by the Speaker: John Urschel is a fourth year PhD student in mathematics at MIT. Urschel received both his bachelor’s and master’s degrees in mathematics from Penn State University. In 2017, Urschel was named to Forbes’ “30 under 30” list of outstanding young scientists. His research interests include numerical analysis, graph theory, and machine learning. He is expected to graduate from MIT in Spring 2021.