[petsc-users] snes fieldsplit for kkt

[Tobin Isaac]

On Sun, Mar 24, 2013 at 11:03:20PM -0500, Jed Brown wrote:
> Tobin Isaac <tisaac at ices.utexas.edu> writes:
> 
> > If I'm solving a PDE-constrained optimization problem, I'd like to be
> > able to specify the equations and then choose at runtime between a
> > full-space approach and a reduced-space approach.  Is this possible
> > with petsc?
> 
> PETSc is not an optimization package per se, so you'll have to do a
> little work, at least if your problem is nonlinear or if the size of the
> full space is potentially too large to store. In the specific case of a
> quadratic optimization problem, a reduced space method is equivalent to
> iterating on the reduced Hessian, in which case you can use Schur
> complement reduction via -pc_fieldsplit_type schur. Note that the Schur
> implementation naturally works with 2x2 systems, so you would lump state
> and adjoint equations into the first "block", and then (typically)
> decompose this further in the inner solve.
> 
> George used to have a library called Veltisto that was built on PETSc
> and supported both full-space (LNKS) and reduced-space methods. As far
> as I know, this package is no longer available.
> 
> I think that having a unified interface to reduced-space and full-space
> methods would be very useful to the community. I think that choosing an
> interface for inequality constraints is somewhat challenging because the
> more general forms give the library much less to work with. I know your
> group has quite a bit of experience with this. If you would be
> interested in working with us to put this into library form, I would be
> happy to help.
> 
A unified interface that encompasses inequality constraints is more
ambitious than what I had in mind for now.  What I meant when I said
"specify the equations" was specify the first order optimality
conditions for a problem with pure equality constraints.  Choosing
between full-space and reduced-space would then be just choosing
between snes strategies.