[petsc-users] Tao iterations

[Justin Chang]

Jed,

Thank you for your response. I agree completely with all that you said. I
just wonder what would happen if I attempted to use the TAO routines after
forming the Jacobian J and residual r arising from the advection diffusion
equation.

In my current (linear) diffusion framework, i have the following objective
function and gradient vector:

f = \frac{1}{2} x \cdot J*x + x\cdot[r - J*x^(0)]
g = J*[x - x^(0)] + r

where x is the solution at a given tao iterate and x^(0) is the initial
guess used to compute J and r. These are built upon the assumption that the
solver would converge in exactly one newton step if I used SNESSolve (i.e.,
no unexpected nonlinearities). This methodology works well if I am only
looking at anisotropic diffusion.

With TRON, I would set the Hessian as J, and I imagine the solver would
immediately fail due to lack of symmetry. I am not exactly sure what's
going on within the framework of BLMVM, but given the above objective and
gradient routines, how would BLMVM know whether my equations have an
associated objective functional? Would the solver simply blow up or will it
commit a variational crime by giving me some solution that may not actually
be a global minimum? Normally I would experiment with this myself but
solving something like advection-diffusion using DMPlex isn't trivial when
dealing with high advection to diffusion ratios.

Thanks,
Justin

On Mon, Jun 8, 2015 at 9:39 AM, Jed Brown <jed at jedbrown.org> wrote:

> Justin Chang <jychang48 at gmail.com> writes:
>
> > Hi Jason,
> >
> > One more question about BLMVM... if it only uses gradient information and
> > does not require the definitition of a Hessian Matrix, can this method be
> > applied to solve problems that are nonsymmetric by nature? (e.g.,
> > advection-diffusion equations).
>
> Such equations do not have an associated objective functional, thus you
> don't have the "gradient" of something, you just have a system of
> nonlinear equations.  (There are ways to formulate such systems as
> optimization, but they have issues like you mention.)  Reformulating
> nonlinear equations as optimization can also turn a problem with a
> unique solution into one with local minima, for which it may be
> impossible to guarantee that you have reached a global minimum.
>
> Use SNES for solving nonlinear equations.  You can try the quasi-Newton
> methods, which are related to BLMVM.
>
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