[petsc-users] Implementing a homotopy solver

[Lawrence Mitchell]

> On 6 Jul 2018, at 17:30, zakaryah <zakaryah at gmail.com> wrote:
> 
> ​Thanks for your help, Barry.
> 
> I agree about the preconditioning.  I still don't understand why I don't need a particular solver for my shell matrix.  My reasoning is that KSP is easy with M but difficult with A, since A has a dense row and column, whereas M is entirely sparse.  Sherman-Morrison seems to be an efficient way of dealing with this but I could be wrong.​

As Barry says, low-rank perturbations are not terrible for the convergence of Krylov methods.  E.g. 

Theorem 2.1 of https://arxiv.org/pdf/1508.07633.pdf

and a refinement in https://arxiv.org/pdf/1612.08369.pdf with improved bounds.

Cheers,

Lawrence