[petsc-users] Implementing a homotopy solver

[zakaryah]

Thanks Barry and Lawrence - this makes sense now.  MatMult for the shell
matrix should take care of everything - I'll do some tests on
convergence/performance.  Thanks!

On Fri, Jul 6, 2018 at 12:44 PM, Lawrence Mitchell <
lawrence.mitchell at imperial.ac.uk> wrote:

>
> > 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
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