Theory of Krylov subspace methods
and its relationship to other mathematical disciplines

About the Project

Theory of Krylov subspace methods and its relationship to other mathematical disciplines is a project of the Grant Agency of the Academy of Sciences of the Czech Republic, identification code IAA100300802. The project has started on January 1st, 2008 and is planned for 5 years.

Keywords

Theory of Krylov subspace methods, numerical linear algebra, Gauss quadrature, model reduction, matching moments, convergence, stability, complexity.

Aim of the Project

We propose to continue our work in theory of Krylov subspace methods which has in recent years led to revealing of unexpected relationships and results in seemingly unrelated areas. We consider the links between different disciplines a very powerful tool and wish to exploit them further in our work. We will focus on hard questions and plan to investigate in particular: sensitivity of the Gauss-Christoffel quadrature and its relationship to the CG and Lanczos methods, extension of the core problem theory to multidimensional case, numerical stability and stopping criteria in iterative solvers, convergence behaviour in relation to various properties of the problem and open questions in the mathematical theory of optimal short term recurrences.

International collaboration

During the project we plan to continue long lasting and very rewarding collaboration with coauthors of our papers, in particular with Mario Arioli, RAL, England; Michele Benzi, Emory University, USA; Ake Bjorck, Linkoping University, Sweden; Vance Faber, BD Biosciences, USA; Luc Giraud, ENSEEIHT, France; Anne Greenbaum, University of Washington, USA; Martin Gutknecht, ETH Zurich, Switzerland; Julien Langou, University of Colorado at Denver and Health Sciences Center, USA; Jorg Liesen, TU-Berlin, Germany; Gerard Meurant, CEA, France; Jim Nagy, Emory University, USA; Dianne P. O’Leary, University of Maryland, USA; Chris C. Paige, McGill University, Canada; Valeria Simoncini, University of Bologna, Italy; Martin Vohralik, Laboratoire Jacques-Louis Lions, France; Pavel Jiranek, CERFACS, France.

In addition to that we hope to intensify collaboration with Daniel Szyld, Temple University, USA (on numerical stability and inexact Krylov subspace methods); We will seek advice from and regularly communicate with many other leading researchers in the field, including Michael Eiermann and Oliver Ernst from TU Bergakademie Freiberg, Germany; Volker Mehrmann, TU Berlin, Germany; Per Christian Hansen, TU of Denmark, Sabine Van Huffel and Diana Sima, KU Leuven, Belgium. We have fruitful and intensive exchange of ideas with many other researchers in the field, and it can lead to a joint work in the future.