2022 Lecturer: Anne Greenbaum
Two of my Favorite Problems
I will discuss two problems on which I have worked extensively and the many remaining open questions.
The first involves the Lanczos algorithm for constructing an orthonormal basis for the Krylov space corresponding to a Hermitian matrix A and a given vector b. The vectors produced can be used to solve linear systems, compute eigenvalues/vectors, evaluate matrix functions f(A)b, etc. And while this algorithm is very widely used, it is, in the most intuitive sense, dramatically unstable. Behavior of the best implementations using the best computer arithmetic have been explained to some extent, but with implementations being developed for single and half precision and with computations being rearranged to make better use of parallelism, etc., it is important to know what will and will not work.
The second has to do with nonsymmetric matrices and operators. It is known that eigenvalues alone provide no information about the behavior of Krylov space methods such as GMRES; the field of values, or, numerical range provides some information, but it is too large a set. I will discuss K-spectral sets and what they can tell us about the behavior of Krylov space methods and other problems in numerical analysis.
Professor Anne Greenbaum has had a long-lasting and significant impact on many aspects of numerical linear algebra. She is an expert in the mathematical behavior of iterative methods and has solved many fundamental problems in convergence theory for linear systems and eigenvalue problems, non-normal matrices and functions of matrices. Greenbaum is the author of highly respected books on the subject, including “Methods for Solving Linear Systems,” published by SIAM, and (with Tim Chartier) “Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms,” published by Princeton University Press. She was also one of the original developers of the LAPACK software that has been a workhorse of scientific computing for several decades. Greenbaum is a dedicated and effective teacher and mentor.
Anne Greenbaum is Professor of Applied Mathematics at the University of Washington. She received her Bachelor’s degree in mathematics from the University of Michigan in 1974. She then landed a job at Lawrence Livermore National Laboratory and, shortly thereafter, she began a doctoral program at the University of California, Berkeley. She received her PhD from Berkeley in 1981. In 1986, she accepted a research position at the Courant Institute, where she stayed until 1997, when she came to the University of Washington as a Professor in the Mathematics Department. In 2009, she moved to the Applied Mathematics Department. In 2015, she was elected a Fellow of SIAM; other honors include the 1997 B. Bolzano Honorary Medal for Merit in the Mathematical Sciences from the Academy of Sciences of the Czech Republic, and the SIAG Linear Algebra award, joint with Zdeněk Strakoš, for Outstanding Paper in Applicable Linear Algebra during 1991-1993.