Preconditioned Krylov subspace methods for the numerical solution of sparse and large linear systems have come a long way in the past few decades. For some types, such as symmetric positive definite systems, great progress has been made, and in some cases we can converge to a solution in (near) linear complexity. Contrary to that, for symmetric indefinite and nonsymmetric systems, significant difficulties persist. In this talk I will discuss some of these challenges. We will focus on double saddle-point systems, and describe the elusive quest for scalable block preconditioners, some of the merits and weaknesses of spectral analysis, and the promise as well as the limitations of field-of-values analysis.

Refreshments will be served before the talk, starting at 2:45.