Matrix iterative methods from the historical, analytic, application, and computational perspective

Prof. Dr. Zdenek Strakos

Abstract

Modern matrix iterative methods represented, in particular, by Krylov subspace methods, are fascinating mathematical objects that integrate many lines of thought and are linked with hard theoretical problems.

Krylov subspace methods can be seen as highly nonlinear model reduction that can be very efficient in some cases and not easy to handle in others. Convergence behaviour is well understood for the self-adjoint and normal operators (matrices), where we can conveniently rely on the spectral decomposition. That does not have a parallel in non-normal cases. Theoretical analysis of efficient preconditioners is therefore complicated and it is often based on a simplified view to Krylov subspace methods as linear contractions. In numerical solution of boundary value problems, e.g., the infinite dimensional formulation, discretization, and algebraic iteration (including preconditioning) should be tightly linked to each other. Computational efficiency requires accurate, reliable and cheap a posteriori error estimators that relate the discretization and algebraic errors in order to construct an appropriate (problem dependent) stopping criteria. Understanding numerical stability issues is crucial and this becomes even more urgent with increased parallelism where the communication cost becomes a prohibitive factor.

The presentation will concentrate on ideas and connections between them with emphasizing the historical perspective. Technical details will be kept at minimum so that the lecture is accessible to a wide audience.

The presentation will benefit from the material present in the recent monographs coauthored with Josef Malek [Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs, SIAM Spotlights, SIAM, Philadelphia, 2015, http://bookstore.siam.org/productimage.php?product_id=579], and with Jorg Liesen [Krylov Subspace Methods, Principles and Analysis, Oxford University Press, Oxford, 2013, https://global.oup.com/academic/product/krylov-subspace-methods-9780199655410?cc=cz&lang=en&], as well as from several recent papers with Martin Vohralik, Jan Papez and Tomas Gergelits.

Registration

No registration required.

Date and Place

All sessions take place in Seminar Room 02.06.011

• Tuesday, 22.11.2016:
  • 12:15 - 13:45

• Wednesday, 23.11.2016:
  • 12:15 - 13:45

• Thursday, 24.11.2016:
  • 10:15 - 11:45 AND 16:15 - 17:45

• Friday, 25.11.2016:
  • 12:15 - 13:45