syfi team mailing list archive

[Jed Brown <jed@xxxxxxxx>]

On Sat 2009-04-18 19:43, kent-and@xxxxxxxxx wrote:
> > On Sat 2009-04-18 17:32, kent-and@xxxxxxxxx wrote:
> >> For the above equation I think there should be only one minimum and so
> >> it will head in the right direction. I'm not sure whether it should be
> >> quadratic.
> >> If it is convex it can have several minium and it may therefore head
> >         ^^^^^^^^^
> >
> > You probably mean "is not convex".
> >
> >> towards different minima at different times but I didn't think it could
> >> head
> >> off to infinity (or creating a singular matrix) ?
> >
> > Convexity is not sufficient to guarantee global convergence, you need a
> > minimum to exist (consider exp(x)) and the Jacobian should be Lipshitz
> > continuous with J^{-1} bounded.  These conditions are probably satisfied
> > for the models you are discussing.
> >
> > Jed
> 
> Right, but isn't there some results that say that if the functional F(u) goes
> to infinity when u goes to +/- infinity and the functional is convex.

I was probably being overly pedantic, but there is something here
(although it might not apply to your problem).  If F is convex and F \to
\infty as |u| \to \infty, there is a unique minimum u^*.  This really is
not enough to for ||J^{-1}|| to be bounded.  Consider minimizing the
convex functional

  x^2, x<1
  1+x, x>1

If you start at x=2, J is singular.  J is also not Lipschitz continuous,
but mollification would make it so, boundedness of ||J^{-1}|| is still
needed.

> The SVK and Fung demos do still not work properly, but it might be that we
> start with functions to far away. Or there might be a bug...

Is the Jacobian correct?  You can check this by seeing if the Krylov
iteration with matrix-free Jacobian and LU preconditioner converges in
one iteration (-snes_mf_operator -pc_type lu in PETSc).  If the Jacobian
is correct but residuals increase, you might need a line search.  I
think ||J^{-1}|| is bounded for these problems so I don't think line
search will fail.

Jed

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