Speaker
Description
Taking the 1952 landmark paper of Hestenes and Stiefel on the conjugate gradient method as their historical starting point, Krylov subspace methods for solving linear algebraic systems have been around for more than 70 years. Tens of thousands of research articles on Krylov subspace methods and their applications have been published by authors coming from the most diverse scientific backgrounds. Nevertheless, many questions about the behavior of Krylov subspace methods both in exact arithmetic and finite precision computations remain open.
In this talk we will present several examples from the recent paper [1] that illustrate important and practically relevant open questions about Krylov subspace methods. We will focus on the nontrivial nonlinear behavior of the methods, which is their main mathematical asset as well as their beauty. Our main goal is to argue that, despite their long history and widespread use in practical applications, Krylov subspace methods should still be seen as mathematical objects that are worth studying. Any progress in their understanding, even of their mathematical fundamentals, will bring us a step further in exploiting their full nonlinear computational potential.
The talk will be based on joint work with Erin Carson and Zdenek Strakos (Charles University, Prague).
[1] E. Carson, J. Liesen and Z. Strakos, Towards understanding CG and GMRES through examples, Linear Algebra and its Applications, 692 (2024), pp. 241-291.
