Overdetermined, non-linear

[Matthew Knepley]

On Feb 5, 2008 3:26 AM, Erlend Pedersen :.
<erlend.pedersen at holberger.com> wrote:
>
> On Sun, 2008-02-03 at 19:59 -0600, Matthew Knepley wrote:
> > On Feb 1, 2008 5:54 AM, Erlend Pedersen :.
> > <erlend.pedersen at holberger.com> wrote:
> > > I am attempting to use the PETSc nonlinear solver on an overdetermined
> > > system of non-linear equations. Hence, the Jacobian is not square, and
> > > so far we have unfortunately not succeeded with any combination of snes,
> > > ksp and pc.
> > >
> > > Could you confirm that snes actually works for overdetermined systems,
> > > and if so, is there an application example we could look at in order to
> > > make sure there is nothing wrong with our test-setup?
> > >
> > > We have previously used the MINPACK routine LMDER very successfully, but
> > > for our current problem sizes we rely on the use of sparse matrix
> > > representations and parallel architectures. PETSc's abstractions and
> > > automatic MPI makes this system very attractive for us, and we have
> > > already used the PETSc LSQR solver with great success.
> >
> > So in the sense that SNES is really just an iteration with an embedded solve,
> > yes it can solve non-square nonlinear systems. However, the user has to
> > understand what is meant by the Function and Jacobian evaluation methods.
> > I suggest implementing the simplest algorithm for non-square systems:
> >
> > http://en.wikipedia.org/wiki/Gauss-Newton_algorithm
> >
> > By implement, I mean your Function and Jacobian methods should return the
> > correct terms. I believe the reason you have not seen convergence is that
> > the result of the solve does not "mean" the correct thing for the iteration
> > in your current setup.
> >
> >    Matt
>
> Thanks. Good to know that I should be able to get a working setup. Are
> there by any chance any code examples that I could use to clue myself in
> on how to transform my m equations of n unknonwns into a correct
> function for the Gauss-Newton algorithm?

We do not have any nonlinear least-squares examples, unfortunately. At that
point, most users have gone over to formulating their problem directly as
an optimization problem (which allows more flexibility than least squares) and
have moved to TAO (http://www-unix.mcs.anl.gov/tao/) which does have
examples, I believe, for optimization of this kind.

If you know that you only ever want to do least squares, and you want to solve
the biggest, parallel problems, than stick with PETSc and build a nice
Gauss-Newton
(or Levenberg-Marquadt) solver. However, if you really want to solve a more
general optimization problem, I recommend reformulating it now and moving
to TAO. It is at least worth reading up on it.

  Thanks,

     Matt

> - Erlend :.
-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which
their experiments lead.
-- Norbert Wiener