Center for Applied Mathematics Colloquium
The solution of large symmetric eigenvalue problems is central to applications ranging from electronic structure calculations to the study of vibrations in mechanical systems. A few of these applications require the computation of a large number of eigenvalues and associated eigenvectors. For example, when dealing with excited states in quantum mechanics, it is not uncommon to seek a few tens of thousands of eigenvalues of matrices of sizes in the tens of millions. In such situations it is imperative to resort to `spectrum slicing' strategies, i.e., strategies that extract slices of the spectrum independently. The presentation will discuss a few techniques in this category, namely those based on a combination of filtering (polynomial, rational) and standard projection methods (Lanczos, subspace iteration). Filtering consists of computing eigenvalues and vectors of a matrix of the form B = f(A), where f is typically a polynomial or rational function. With the mapping f the wanted eigenvalues of the original matrix are transformed in such a way that they become easier to extract. This particular area blends ideas from approximation theory with standard matrix algorithms. The presentation will emphasize rational filtering and will discuss some recent work on nonlinear eigenvalue problems.
Yousef Saad is a College of Science and Engineering (CSE) distinguished professor with the department of computer science and engineering at the University of Minnesota. He received the "Doctorat d'Etat" from the university of Grenoble (France) in 1983. He joined the university of Minnesota in 1990 as a Professor of computer science and a Fellow of the Minnesota Supercomputer Institute. He was head of the department of Computer Science and Engineering from January 1997 to June 2000, and became a CSE distinguished professor in 2005. From 1981 to 1990, he held positions at the University of California at Berkeley, Yale, the University of Illinois, and the Research Institute for Advanced Computer Science (RIACS). His current research interests include: numerical linear algebra, sparse matrix computations, iterative methods, parallel computing, numerical methods for electronic structure, and linear algebra methods in data mining. He is the author of two monographs and over 190 journal articles. He is also the developer or co-developer of several software packages for solving sparse linear systems of equations and eigenvalue problems including SPARSKIT, pARMS, ITSOL, and EVSL. Yousef Saad is a SIAM fellow (class of 2010) and a fellow of the AAAS (2011).