Iterative Methods in Computational Mathematics:
Analysis, Preconditioning, and Applications

About the Project

Iterative Methods in Computational Mathematics: Analysis, Preconditioning, and Applications is a project of the Grant Agency of the Czech Republic, identification code 13-06684S. The project has started on February 1st, 2013 and is planned for 5 years.

Keywords

Krylov subspace methods, preconditioning, convergence analysis, regularization methods, total least squares, sparse matrices.

Aim of the Project

The project deals with iterative methods for several important problems of numerical linear algebra. It includes their basic phases starting from their analysis, involving preconditioning, solving ill-posed problems as well as real-world applications. We focus on Krylov subspace methods, open questions related to their convergence, associated matrix approximation problems, error estimation and stopping criteria. We study various preconditioning techniques including new algorithms based on incomplete factorizations and orthogonalization schemes, and block saddle-point preconditioning. The project deals also with analysis of regularization methods for solving ill-posed problems in image and signal processing, open problems in total least squares and Golub-Kahan bidiagonalization. An inseparable part of our work is a broad international collaboration and selected real-world applications such as the approximation of scattering amplitude and nuclear magnetic resonance.

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; Reijo Kouhia, University of Tampere; Jennifer Scott, Rutherford Appleton Laboratory; Alicja Smoktunowicz, Polish Academy of Sciences. 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 also lead to a joint work in the future.