Re: deal.II workshop: follow-up question

[Wolfgang Bangerth]

Hi Alexander,
let me take the liberty of punting this question to the mailing list.


> this is Alexander Grayver from ETH. I got a question after listening to your
> talk about KINSOL last Friday. I could not stay till the question session and
> could not ask it then. Hopefully it is ok to do it now.
>
>
> The question is whether you think KINSOL would also be
> appropriate/advantageous as a solver for regularizaed non-linear inverse
> problems? I specifically mean problems that can be posed as PDE-constrained
> optimization (say seismic tomography as an example).
>
>
> It does seem that all the internal logic within the KINSOL that helps save
> time and resources can be applied to PDE-constrained optimization problems,
> but maybe there are some important differences/peculiarities that I'm missing.

I see no reason why KINSOL couldn't be used for this: As long as you can write
something as a system of nonlinear equations, KINSOL should work. In your
case, the nonlinear system would be the one that arises from setting the
derivatives of the Lagrangian to zero.

I will say the following, however: KINSOL is just a nonlinear solver, but not
specialized to problems that arise from optimization. Optimization problems
have more structure that good optimization packages can exploit. For example,
it is possible that one could use different step lengths for primal and dual
variables, or exploit that some equations arise from inequality constraints.
One might also want to use interior point methods where one modifies the
penalty parameter in front of the logarithmic penalty terms in each nonlinear
iteration. This is, for example, what Justin O'Connor's new step-79 does, and
it shows that there is a lot of algorithmic work one can do that goes beyond
just solving a nonlinear root-finding problem.

In other words, it's *possible* that specialized optimization packages could
get more out of the problem, but using KINSOL would probably get you 80%
there. In particular, KINSOL has an interface that just fits very nicely into
deal.II-based programs because all it requires is the solution of linear
systems -- something that you have likely already implemented.

Best
W.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/