Linear systems arising from large-scale constrained optimization problems or partial differential equations with constraints frequently appear in a saddle-point block form. The associated matrices are often large, sparse, symmetric, and indefinite. The numerical solution of such systems is a challenging task in numerical linear algebra. If state-of-the-art iterative methods are considered, it is important to design solution techniques that take into account the numerical properties of the underlying discrete operators. In this talk we will provide an overview of such solution methods, with focus on the special situation where the leading block of the matrix is highly rank deficient. This situation arises, for example, in the solution of certain electromagnetics problems. We show that it is possible to utilize null spaces as an alternative to Schur complements. Furthermore, new minimum residual short-recurrence methods can be designed, which is capable of efficiently handling singularity. Numerical experiments validate our analytical observations.
Bio: Chen Greif is Professor and Head of the Department of Computer Science at the University of British Columbia in Vancouver, Canada. His main research area is numerical linear algebra, within the field of scientific computing and numerical analysis. He specializes in preconditioning techniques for iterative methods for solving large and sparse linear systems. Prior to taking on a professorial position with UBC (2002), he was a senior software engineer at Parametric Technology Corporation in San Jose, California (2000-2002) and a postdoctoral fellow at Stanford University (1998-2000). He obtained a PhD in applied mathematics from UBC in 1998. Chen is a co-author (with Dr. Uri Ascher) of the SIAM bestselling book, A First Course in Numerical Methods, and a co-editor of two other books. Chen presently serves as SIAM Secretary, and is a member of the SIAM Council. He is an associate editor of the SIAM Journal on Scientific Computing, among other responsibilities. In 2014-2017 he was Chair of the Gene Golub SIAM Summer School Committee, and he was previously the Program Director of the SIAM Activity Group on Linear Algebra. Among his recent major conference organization roles, Chen was Chair of the Organizing Committee of the latest International Conference on Preconditioning Techniques for Scientific and Industrial Applications (2017) and co-Chair of the SIAM Conference on Applied Linear Algebra (2015).