Re: [freeeos-general] Status report for the BFGS effort

[Alan W. I. <ir...@be...>]

On 2006-12-19 19:51-0800 Alan W. Irwin wrote:

> So I am now in the middle of doing a "final" BFGS implementation[....]

> [...]for fixed input auxiliary variables, iterate
> just on fl for consistency between the calculated and input rho. In this
> way, fl is implicitly eliminated in terms of the input rho so the solution
> of the EOS and in particular the number densities become implicit functions
> of just the input auxiliary variables for fixed (input) rho, T, and
> abundance.  Under these special conditions (and only these conditions) with
> fl implicitly eliminated, it is straightforward to show that minimizing the
> free energy as a function of auxiliary variables is equivalent to minimizing
> the free energy as a function of number densities which (see Paper II) is
> exactly equivalent to the solution of the EOS using the Newton-Raphson
> technique.

More progress....  Before I had implemented the ideal component of the
free-energy using a formula that was only correct in chemical equilibrium
(since it depended on equilibrium conditions being true).  Now I have
changed that formulation to the fundamental one (see MHD, Paper II). The
other issue was that before some components of the free-energy formulation
were evaluated at the "old" number densities, while others were evaluated at
the "new" number densities of the N-R iteration.  Now, all components are
evaluated consistently with the "new" number densities.  Those in turn are a
function of the degeneracy parameter, fl, and the "old" auxiliary variables
of the N-R iteration.  These two changes now meet the theoretical conditions
that are required to minimize the free-energy as a function of auxiliary
variables (assuming fl has implicitly been eliminated in terms of the input
rho).

I am extremely happy to say that the numerical result is just what was
theoretically predicted above.  The free-energy (for implicitly eliminated
fl) is minimized (actually quartically) by the Newton-Raphson convergence
and is equal to the free-energy calculated by equilibrium relations at N-R
convergence as expected since N-R convergence means chemical equilibrium has
been obtained.

So I now have a proof of concept that the "final" BFGS implementation is
going to work. The next steps are to (1) implement a calculation of the
gradient of the free-energy with respect to the auxiliary variables in this
formulation (for fixed input rho, T, and abundance), and (2) use the
free-energy and its gradient to minimize the free-energy using the BFGS
technique when the existing Newton-Raphson technique is failing to converge.
So there is still a long way to go, but I am happy the proof of concept
finally worked on the last day in 2006, and I am hoping for a good BFGS
result early in 2007 as a result of this breakthrough.

Happy New Year, everybody!

Alan
__________________________
Alan W. Irwin

Astronomical research affiliation with Department of Physics and Astronomy,
University of Victoria (astrowww.phys.uvic.ca).

Programming affiliations with the FreeEOS equation-of-state implementation
for stellar interiors (freeeos.sf.net); PLplot scientific plotting software
package (plplot.org); the Yorick front-end to PLplot (yplot.sf.net); the
Loads of Linux Links project (loll.sf.net); and the Linux Brochure Project
(lbproject.sf.net).
__________________________

Linux-powered Science
__________________________