On preconditioner updates for sequences of saddle-point linear systems

Updating preconditioners for the solution of sequences of large and sparse saddle-point linear systems via Krylov methods has received increasing attention in the last few years, because it allows to reduce the cost of preconditioning while keeping the efficiency of the overall solution process. This paper provides a short survey of the two approaches proposed in the literature for this problem: updating the factors of a preconditioner available in a block $LDL^T$ form, and updating a preconditioner via a limited-memory technique inspired by quasi-Newton methods.