The Bramble-Pasciak Conjugate Gradient algorithm is a widely used tool in the finite element community. Motivated by a reformulation of the linear system in saddle point form, we introduce Bramble-Pasciak-like methods that can be used to solve problems coming from optimization. We illustrate that the eigenvalues for the preconditioned matrix in this setup have a very similar (sometimes equivalent) structure to the preconditioned matrix of a method which uses a constraint preconditioner. We furthermore give numerical results for optimization examples.
The author would like to thank Sue Dollar, Nick Gould and Andy Wathen for their helpful comments.