Projected Krylov Methods for Saddle-Point Systems

Projected Krylov methods are full-space formulations of Krylov methods that take place in a nullspace. Provided projections into the nullspace can be computed accurately, those methods only require products between an operator and vectors lying in the nullspace. We provide systematic principles for obtaining the projected form of any well-defined Krylov method. Projected Krylov methods are mathematically equivalent to constraint-preconditioned Krylov methods provided the initial guess is well chosen, but require less memory. As a consequence, there are situations where certain known methods such as MINRES and SYMMLQ are well defined in the presence of an indefinite preconditioner.